Buddy, back today with my sisters on the battlefield. Literally, my stalactic sisters on the battlefield, Taylor and Stephanie and my, oh my, what a month it's been, ladies.  Like, or both, maybe it's both, I don't know. Both, yeah, both. I'm writing a book on this one day, definitely. Oh yeah, and I know you guys, and I said this in my last follow-up video, it's the first video we did together, which thank you guys. Those videos are going out like crazy. I typically don't watch the numbers on shows I do. I just kind of put my stuff out there and that's it. I always feel like the right people are gonna see it, but those videos are getting like really pushed out, but we know that where we are on the internet is the actual battlefield right now. So it doesn't surprise me because obviously we have the help from our galactic family and yeah, which we will definitely have to talk more about one day when we're able to because that has been probably one of the most exciting and surprising elements of this full great awakening that I personally have experienced is starting to have that inner type of relationship with the off-worlders and how powerful they are. And so, but we'll get more into that after, the battle is done because as we know we are in a battle so we can't say everything upfront and out front and in the open, but one thing we wanted to really focus on today guys and I know that I've gotten a lot of emails and a lot of comments regarding who this, what entity is and we're not gonna say who it is right now guys for lots of reasons. It's just not appropriate right now to say it on the platform and we are in the middle of a legitimate WAR and so we have to be very careful and very strategic about things. But the one thing that's bothered me the most personally is because I kinda wanna be like you guys, the whole point of the great awakening is trusting your gut. You don't need us to tell you who it is. You literally don't need us to tell you who it is. You will know who it is by your gut reaction to this particular person. So let's get into that ladies. Let's talk about this whole, we've heard this so many times during this great awakening, right? Like trust your gut, trust your gut, trust your gut. But then we also want to have like confirmation from another person, but we don't need that, do we? The inner knowing is so strong and that's the thing is like, a lot of you guys, since you guys were children, since you guys woke up, whatever that may be for you, we call it the compass of the heart center. You probably instantly had these feelings of being like attracted to someone or drawn away from someone. And that's how we want you guys to start navigating to is by that compass of the heart center. Does this feel good? It doesn't even have to be a person, it could be what you're doing too. Does this feel good? Does this not feel good? And we're gonna call it, we're instead of listening for truth because people can say really beautiful things. We're gonna call it feeling for truth. The feeling for truth, trust in your gut, your heart center and your intuition. It's just a different way of being, but you guys have always done it, right? Yeah.  And we learned to like kind of strip people of having that faith in what they know and hear. And I love how you say that, like leaning on your feelings and not words because people can speak real pretty to you. They can speak real pretty, that's spell casting, hello. They can speak real pretty to you, but your gut always knows. And that's where we get this idea of cognitive dissonance, which is a lot of people, even in the normal society, understand what cognitive dissonance is, is when you're hearing one thing, seeing and feeling another, but you believe what you're hearing, you're believing the lie over what you actually know to be the truth. We see this a lot with CULTs, you know, we, I know, sorry guys, you know, we're still under censorship. We have to watch certain words, but let's talk about that more too, because people do often will second guess themselves for what somebody says to them, right? Which spell casting is a form of manipulation and that manipulation comes through mostly through words. Mm-hmm, you know? And I'll reiterate too, I talked about this on our last week's video for me when I saw this particular entity. I literally got nauseous, sick to my stomach. I felt like a gut punch. I had a literal like physical reaction to it.  What are the words, how do the words feel? Physically, mentally, emotionally, how does it feel? And does their face match what they're saying? Does their body language? I mean, these are all things we learn throughout our lives is you can tell when someone's genuinely happier, when someone's also not genuinely happier being genuine as well. Energy doesn't lie. No, when they're faking it, you can tell right away. It's like people always say that a used car salesman is the best actor ever, but he's not because you know he's acting. Yeah. He's lying. That's good. I know that, right? Right? Well, it's also, so what do we see a lot with like spell casting or even just dealing with the narcissist? We see what we call love bombing. That's an actual term. And it's a term of manipulation.  And that's a form of spell casting and manipulation. What they're doing is putting you under their control because now you become their source of, or they become your source of validation, right? And I know so many people who've gotten into relationships including myself that started off with love bombing and after the relationship is over, they knew when they first were getting love bomb that something wasn't right. Something wasn't right. We also see with manipulation that there is this tactic called a mirroring. There are things that now kids do mirror. So when a child is growing up, they learn to mirror things from their parents and their siblings as part of development. So they learn things. But and there could be when you meet friends, this can happen with friends or a boyfriend or girlfriend, whatever, you're obviously gonna have a lot in common. There's obviously gonna be a lot of things that you have in common. But with mirroring, when that person tries to put themselves in your direct eyesight with no one else where they're completely mimicking you to the point where you think you've met your other half, that's manipulation. Even if you do have your other half, your twin flame, there's still gonna be some differences between you two. And a healthy relationship should not have to worry about the fact that you might like to drink rum or they might like to drink whiskey. Something as minor as that. You know what I'm saying? So that in itself is, this is literally, even though we're talking about psychological stuff, this is literally all part of skill casting. And putting someone under their spell. Well, what happens after the love bombing when they've got you where they want you, then comes the degradation. They pull you. Like a hook, isn't it?  And remember, like nobody should have power over you. No one should. And the biggest way to take your power back, and this took a lot of us as impasse and I'm sure a lot of people are impasse. And, you know, we feel this feeling that we have to give and give and give because we're emotional and we feel like, that's what we're supposed to do. Well, yeah, we are supposed to give and be service to others. We have to be so full that we're not draining our energy. And I always say this to people, the biggest form of self-care, the biggest form of taking your power back and saying no to people, places and things that do not serve you. And it's not an equal exchange of energy at all. You're drained. Yeah, yeah. And it's also, you know, I was thinking about this too, like in a healthy dynamic. Now, I'm sure that you guys all have friends like we pretty much have a text chain goat. The three of us have a text chain that's pretty much nonstop going. My best friend's the same way.  But if one of you were to stop responding for a couple of hours, that's fine. You're probably doing something, right? Normal people. Always me. Well, yeah, I go to the bathroom. I'm like, guys, and I'm like putting my phone, I'm like, they're gonna have so much fun without me. It's almost fear of missing out.  So like if you meet someone and you have initial feelings about them, you're gonna be able in a healthy environment to step away from that and settle into your feelings. But if someone is trying to manipulate you and spell, they're not gonna give you that time. They're gonna put you in a place of confusion so that you're not actually aware. And I know that narcissists themselves have a way of manipulating your gut feelings too. That's why it's important to be able to have a sense of like boundaries and a sense to be able. And normal, healthy people aren't gonna respect those boundaries. And if you say, hey, I gotta go home for a bit, they're gonna be like, that's cool. Talk to you soon, you know? And let you have that time. And I love how you talked about the sense of self as well because I know that with a lot of like, now of course the entity we're dealing with is, we're not even sure if it's human. So this applies to mostly humans. When you don't have a sense of self, you have nothing to fall back into like a sense of who you are, a sense of self. You will try to then take that essence from other people. And that's what narcissists do. That's why they love bomb. That's why they hook people. That's why they literally feed off people.  And then when you can't make them happy, when they are done getting their narc supplier, their sense of self from you, then that's when they start degrading you. Because all of a sudden, now you've made them feel worthless because they didn't have a sense of self anyway. And even though that's dealing with just the humankind, we can kind of see that in a greater extent with like spellcasting. As I said, we've now learned, stealing natal charts apparently is quite common when it comes to black magic. Until recently myself, yeah, I didn't know. How many people do we believe have had their natal charts stolen in this whole? What number did we get? I think it was like 34. So I'm one of 34 people who's had their natal chart manipulated. And we know of a few others. Multiple people. Multiple people. That are involved. This is bigger than just me.  So we're definitely talking about some high ranking black magic dark cult members. Now, a normal person's never gonna wanna do that anyway. Mess with somebody. No matter how mad at you are at someone, you're never gonna wanna go and like mess with their Akashic record. I think most of us understand that those are sacred. You don't mess with that. So, I mean wild, right? So we know that this is a lot bigger in that, but if you think about the idea of the sense of self, and I think I said this in our last episode, the reason why this entity and the Legion, because we keep getting it to Legion. Yes. My situation is just one of many. It's a micro to a macro of what's happening in this battle on planet Earth right now. Because they are not of the light, they don't have a sense of self. They don't have a human personality. And so they have no choice.  And they are smart enough to then plot and plan on the battlefield of who they're gonna attack because of the domino effect it has with other people as well. And can we explain that a little bit to people? Like how our souls are so intertwined with each other? Like how our natal charts kind of intertwine and our Akashic records kind of intertwine together because I know people kind of ask me about that. Do you ladies have a way of explaining that? Like how soul families work? That's a complicated question. I just, I think it's like, and I agree, I think that's a very complicated question, but I also think it's just like remembering of how many lives you've probably lived and how much you've been through and remembering. So like for example, if you look at how many times you've incarnated on Earth or Off Earth depending on who the person is, remember that you have soul families with each of these collectives. Not only are you just having soul families over here, you also have them over here. So it's almost like every one of us is connected to each other in some sort of way. Remembering, of course, that's unity consciousness too. We're all connected to source. But also remembering that like, we are so intertwined with everyone's information, everyone's stuff, and that's why we can feel it so much too. And that's why you're drawn to people. Or you could also be karma while you're not drawn to people. But at the end of the day, there's a difference because with the spell casting and with a narcissist, I was thinking about this whenever we're talking about source. So all of us get all of our energy straight from source, straight from God creator, straight to source, and that's how we get our information. There's extensions of that in different realms and levels that we can tap into with our soul families. But at the end of the day, these beings, they don't have an energy source. So they use someone else as an energy source. But if we go back to narcissism, it's actually very similar because they're using your energy and your emotions as their energy source versus source itself. And I believe narcissists can probably connect to source if they need to. There's always redemption and learning and going through that. But at the end of the day, if you're getting your energy from anything other than source and other people, then that's leechy. That's a very leechy parasitic thing. And that's why we're talking about the spell casting too because they're literally taking someone's energy and life force in order to get the spell cast going too and to bind people. And there's a lot of interconnectedness like Bryce was saying too. We are so connected, so connected. And you feel it, like you actually feel it when somebody that you, and it's wild, we were just actually talking about this because, and I actually spoke with Tamara this morning, we were gonna film, but we're gonna put it off for next week because we had some stuff to talk about regarding this issue because of what's going on. But we are so connected to each other. And in this lifetime, we're now meeting these people, these soul families later in our lives. And even though there are people we're connecting with in this battle, and I'm sure the people watching have experience as well that we haven't known as long as other people in our lives, we feel more comfortable and more connected to, I mean, I think the first time we guys zoomed together, I was like folding my, I almost went folding my laundry last night too, but I was like folding my underwear in front of you ladies. And I was like, hey, what's up? You know, like, good to see you. Fold my bra. And that's, you just have that inner knowing that you know these people. Like you've done this, but we, this is not our first rodeo. You know, we've done this together and we trust each other. Like there's a deep inner trust. And so what starts to happen is when you, I know I think it's in Chinese culture, they talk about these like tentacles where you have this like tentacle attached to people that you are connected to on a very deep spiritual level. And so you think about that when somebody that you were that connected to is going through hell because of spell casting, you actually feel it too. You feel it as well. I just saw an infinity symbol. So yeah, I would say that the interconnectedness of the energy, I would totally go both ways with that exchange of the cord. Remember we have negative cords, but we also have cords to people that we love and we care about. And that's, that's cool. I didn't know bad about Chinese culture, but that makes sense. Cause it looks like a cord to me. Yeah, well that's how you think about it. We call it telepathy. We call it, you know, psychic intuitive knowing, but you think about, have you ever like been driving even before or, you know, I guess long before there were cell phones like at home and, you know, you were like thinking about someone else in the phone rings and they're calling you. Well that would be, could be just described between that energetic cord that you have with that person. It's like how parents like know something's wrong with their child.  There are negative cords, but they're also very positive ones, very, very positive ones. And, and that is why this makes it so complicated. And that's also why we can't go in too much detail about the person or what's happening because there are people involved that we really genuinely love who are part of our sole family that are victims of this, you know, even though we might not be too happy with what's happening and, and the behavior because of this, we still are fighting for them as we would, would hope our family would fight for us as well if we were in this position too and have that compassion to understand that, that they are in a place of total, I mean, that's what we keep getting, right? Total confusion, total, and we're seeing their bodies deteriorate as well, just like mine was deteriorating. We're seeing it physically take, take its toll on these people, but we know that they're gonna be okay. We know it's gonna be okay. And we know that, you know, the devil tried to play us in this battle, but God is actually playing the devil at his own game right now because God used the situation to help some of us wake up and I think he's still gonna continue to use this situation to help other people wake up as to who they really are in the day. And this is one thing I've said before and I, and I've said this privately, but I'm actually gonna say it publicly now. I am not a fan, even though I've laughed about the movie myself, this is all a movie, this is all a movie, and we've used that as a way to describe what's going on because of censorship. It kind of makes me a little bit when people continue to say this is all a movie, this is all movie, this is all movie because yes it is, but we're also in the middle of the biggest W-A-R we've ever been in. And when we think that this is all just, you know, the good guys doing what they have to do, wake people up, we drop our guard and we stop, we lose that sense of spiritual discernment. And I think that what we're living through right now is a good example of that happening, of losing spiritual discernment because maybe the guard was dropped a little bit. And when it comes to spiritual warfare, like you have to, going back to that feeling, you have to trust yourself. And I have the same reaction Stephanie did when I first saw the thumbnail of this person come up, which we know her thumbnail is not actually the person, which is wild. Like it's- It doesn't look anything like the person. The person we can, but it's still the energy behind it. Before I even knew that this was the person using my needle, but I didn't even know at this point my needle chart was being used. I got that same feeling of I have to throw up. Like it was a visceral, it wasn't just like, oh, I don't like this person. It was a visceral, I have to go throw up now. And we've gotten so many people have told us that they have the same reaction to that. So imagine having that strong of a reaction and then being gaslit and told you're wrong. You still have to trust yourself. And sometimes that means walking away from the person that's telling you you're wrong because you have to trust yourself. To not own self be true. You know? So I mean, I know I've had those situations where I will get a bad instinct about a person, but then I will be tricked into like believing them and thought, I mean, have you guys have that before? All the time.  Like I've had so many like gut feelings about something and then like, it might be some bit like when I was in the medical field, I might be like a patient I had that feeling about or one of the nurses or one of the medical assistants say to me, oh, no, no, that person's harmless. Like there was a particular priest that I used to have to room and he gave me the creeps. And he said something to me when he found out I was getting married that I couldn't get out of the room quick enough. Like when he found that I was engaged, he goes, oh, so Stephanie, does your husband know about us? And I'm like, what? What? Oh, he's harmless. Really? Because to me that doesn't seem harmless. That pervy. That's very pervy. And so I've had those moments and then I try to convince myself, oh, just harmless, just harmless. And then down the road I find out, no, that person is not harmless. That person is a perv or a pedo, you know. Yeah, it's funny. The other day we were just zooming privately the three of us and I pulled up my old high school that I went to. And it's funny because I went to a very private school. It was a boarding school and there were, if you look at it, you look at pictures of it, it definitely screams the bad guys basically. And I had asked, there were some faculty members at the school that I never liked as a kid. Like never liked them.  That is God talking to you. That is your protection. That is absolutely, how many people have said like, oh, I knew all along or like, oh man, I should have listened, I knew, you know. I mean, Taylor, I'm sure you've had it before because you've done a lot of work with people, like with the hands on with people and stuff, haven't you? Yeah. Well, it's funny when stuff is talking about it. Mine was always people, but it was also like, I knew when I shouldn't take a patient too, like I was laser hair removal. So I was a little different in the medical aesthetics industry, but I would do laser hair removal and I'd be doing it all day. And I'd be like, I really need a break. This person's really draining me. I would tell that to my boss. I'd be super transparent. And I remember I had trauma from this for a while because this is a repeat cycle in my life where someone, oh, you're just being too sensitive.  No, my intuition's on point, my gut's on point. And that's what it is. But I didn't know that at the time. So you're like, I'm just too sensitive. And you start to believe those things about yourself. And those are not things that you should believe about yourself. You're not too sensitive, you're intuitive.  There's a warning, warning, warning. And it's true. I remember somebody saying, even with love relationships, that we have been taught that in order for us to be in a good relationship, there has to be passion and drama and arguing and all this stuff. And someone was like, no, if that's what you're feeling, that's not a good relationship. A good relationship should be very even keeled. You should not have to, but we see the movies. We see the stories. And so we get ourselves in these situations where we're not listening to what our gut is telling us about situations. And so I really want to just like, if I could shake everybody watching, be like, trust yourself. Trust yourself. I don't want to be your guru. I know Taylor doesn't want to be your guru. Stephanie doesn't want to be your guru. We want to be your sisters. Trust yourself. Listen to what your intuition is telling you. And from the first, what is it they say like you make up your first impression on someone like the first minute of meeting them? You've already got like your first impression of them. That means your personal spidey senses are going off right away. People have auras around them too. People have vibrational frequencies that you can feel physically, mentally, and spiritually. The other thing I want to bring up too is this kind of goes along with the lines of relationships that are narcissistic, but probably fit into this as well is somebody might come off and sweet talk you in and be all nice and then they'll mistreat you. But then you have this period which is called the honeymoon period of the cycle where, oh, I'm so sorry. I'm so sorry. And then they're pouring their love out to you, they're pouring their heart out to you. And they make you feel like it's OK and it's just you. And then the gaslighting then starts, oh, well, they try to manipulate you thinking that they were wrong. It's this whole cycle. And you can actually Google narcissistic relationship cycle and actually see a picture of the cycle. Which is very much as Taylor said, it's very much like spellcasting, what they do is spellcasting. So the bottom line is don't listen to words, listen to feelings. And also know what you're worthy of. I think it took us all a long time to feel like we were worthy. And there's still times that we all are going to struggle with that. There's times that we look at ourselves more like, well, I expect this for myself or my mom expected this for me or someone expected this for me.  You're worthy of all these things. But the hardest thing to learn is how, I always tell people this, go look in the mirror and tell yourself how amazing you are and you'll find that it is hard as shit. The first time you look at yourself in the mirror and you try and say those things to yourself, it's hard to believe it. So I say, if you can't love yourself fully, start nurturing yourself fully. And then shift yourself into that love vibration. And then you're able to be able to deal with what's going on out there as well. But I know that that's hard because you feel like you owe people things, right? Like we get in this blood is thicker than water thing. I owe my mom this. I owe my children this. I owe. But no, you don't owe anyone anything. Half of these relationships you incarnated into, there's a reason they don't feel like your soul family. It's because you have lessons to learn, karma to get rid of. And sometimes they're meant to trigger you to shake and wake you and make you realize how wonderful, big, bold, and beautiful you actually are too. Right. Right. That's absolutely. And that we actually just shot a video with Katherine Edwards this morning and Shanti and Cindy about Saturn Alien Festival, which what we're in right now is Christmas. And how Saturn has been manipulated. But Saturn was always meant to be the shaker up. That's to bring you some hard truths about yourself, the tough love planet. Of course, they inverted it. But we're going to bring it back to that. Sometimes you have to go through the dark night of the soul. In order to learn how incredible you are. So yes, some of those karmic relationships are just that. And let's talk about, because I do want to pull some cards. But when we talk about people, this whole idea of soul mates versus twin flames versus all this kind, can we kind of give a broad definition of who, of what? What's a soul mate, a soul family? What can we tell the audience about those dynamic relationships? Yeah. Well, this is all your thing. Well, I guess for me, it's remembering, too, that like I said before, Bryce was explaining, there are so many intertwined connections. You probably have a lot of soul mates. You have a lot of soul mates. You have a lot of soul family. But you also, for me, a twin flame experience is a very high experience. It doesn't make you better than anyone. All of us are equal. But if you do have a twin flame, normally what it is is two full aspects of one experience. So both of these coming together to bring in their experiences together. Well, twin flame unions, they can be romantic. Absolutely. There are going to be romantic twin flames that are drawn to each other. But sometimes twin flames are just going to work together energetically and coming in together for experience. And we're coming into this huge time. And I'll tell this from a BQH sessions I've been having for the past week. I've noticed a pattern. All of these people are coming into union with people that are like them. And it's about to become physical. And we're already starting to notice that. Calling it the gray gathering like Stephanie was calling it. All of us are going to come physical together. But on top of it, too, I've noticed that there is a huge surprise. Like some people, their akash does not want to let them know what's about to come for them. We had a window open and this beam of light come in for the scalp. And they were like close the curtains on her. They didn't want her to see. And she's like, why can't I see it? Why can't I see it? And it's like the surprise is so big. And I know that sounds cheesy. And I know everyone's really tired. Like I bet a hundred of us are like, well, where's the surprise? Well, it's coming, I promise. And the fact that my BQH sessions are doing this pattern, it's telling me a huge thing's coming for everyone. Everyone. And it's interesting because Stephanie and I had talked about this before because we had noticed this that we're all being rearranged right now. Everybody's being like rearranged right now. We're noticing that dynamics and relationships we thought were one thing or changing to another thing. And we're all being, and I think a lot of people are actually finding their twin flame. I think, and that's part of what this coven is doing is that they're trying to divide that because when you come together, there's power. There's power. There's a reason why this entity did not want Stephanie, Taylor, and me to work together. There's a reason. We don't quite know what that is yet, but they know. We don't. And it's funny because throughout this whole thing, Stephanie and I keep asking Taylor, well, who are we? Like, why are we, I thought we were just cheerleaders on the sideline. Like now someone's got my needle chart. Like what the hell's going on? And Taylor just laughs and said, they keep telling me, don't ruin the surprise. Don't ruin the surprise. Yeah, and it's never been like that. I've never, I hate to be like this guy, but like my information, it doesn't come in like that. Okay, so the fact that I'm getting this all around with a lot of people, it's so much bigger. Like, of course, we're sitting here and experiencing this, but it's so much bigger. And I know Bryce and Stephanie, all your guys is audience too. And I've seen some of the comments. I try not to get too attached or anything, but I do see them and I know for a fact that people are experiencing very similar things to what you guys are experiencing. And it's so important that people know that this is okay and that you're not doing anything wrong. And from a galactic standpoint too, I just want to affirm for you guys, like, yes, like, Dumbs are getting like almost done. Like they're done from what I see, they look done to me, but we have to also remember there's still players on the earth. And we forgot, it's almost like not all of us, but you know, we sometimes forget that the WAR is now physical and they're in our physical realm. But from a galactic standpoint, we knew this all along that we would have to put our swords up for the final battle and it was about us, all of us getting together. And the coolest thing about this is everyone's starting to band together. Yes, because that's kind of what I was saying about the poll. It's just a movie, it's just a movie. Yes, we know that anthropologically we know that we're gonna win, like we know that. And we know, and I will tell you, I've been like almost, I went from being like to the point where I thought I was gonna like lose my life to being absolutely jovial because I now feel, I was just telling them, like I feel so protected. Like I feel my guides around me all the time, even when I'm going to the bathroom, I'm like, you guys really don't need to be in here for this, but I still feel them. And that was that, this whole situation kind of open that door for me to actually understand how protected we are. But, I think it's because we are now realizing that we are the storm, as we said in the first video. We are the battle. This is not just about Mr. T. This is not just about the alliance. We are part of that. We just don't remember. We volunteered as tribute. No, Hargay, we literally- No, no, no, no, no, no, no, no, no, no, no. I will do this. Here we are. And now we've got to do this. And I know you said something, Taylor, once I got me kind of emotional that they wanted to thank us for doing this. There's like this, I know, like I just saw nine of wands in my head. I know guys, like I know how everyone's feeling. I really, really do. Nine of wands is the one where he has an eye patch on him and he's exhausted. I can totally get it, guys. I totally get it. But it's also this celebration is coming. The great gathering is coming, unity is coming. Like I can feel it in my bones and when we look at times, we get, we freak out because I know I've been through this. Like I'll start to feel something or predict something and I'm like, oh my gosh, it's coming. And then all of a sudden, I'm like, it went by and then you get to that like downstage where you're like, well, am I just crazy? Am I just wrong? Just remember that we guys are getting visions and information and stuff. Time is not linear in the quantum.  Like, trust me, I want to date too. I want to know exactly when. But I'm telling you guys, we're in a portal. We're going through the gate. It is almost like the fruition is going to come. And I think you guys would agree with me. If the fruition doesn't come like in the near future, like people are going to be very, very, very tired and very done, like even more than, I know a lot of people already are. I don't mean it that way, but. Yeah, that's what I keep saying too. There's only, you know, even though most of us are learning that we're not fully, like we are star seeds. We do carry the essence of a galactic nature, but we're in a human body and human bodies do get tired. Human spirits get tired. Human minds get tired. And I do believe that we are at the end as well. And I know, I think there is going to come a point, you know, we see these like memes of like activation with the red eyes. I think there's going to come a point where we all just kind of like activate. And I think that, you know, when something so drastic happens, you run the risk of traumatizing somebody. Cause again, we're in a human body, we're in a human brain. And so I do understand that the powers that be on world and off world had to let this unfold in a particular way so that we would be ready. We would, and we were laughing about like, you know, white magic for dummies.  It's not like we had a training day. We were just thrown in, you know, but the more we get thrown in the more I realized like, oh, we've done this before. We have had a training day. We have, and we got God on our side. We got light in us and that light can create. And, and I'm going to say this, we all, I don't think any of us really knew how evil the devil was until we entered this time. How bad, bad truly was. And I'm going to tell you something. Satan's not going to leave. He's not going to bow out gracefully. No.  Shucks. I just saw explosions. I was like, please no. He's not, well, maybe that's the people that are going to be, you know. And also, I think that could just be them throwing like trying to throw a bunch of distortion, seven of cups everywhere too. Like they can look here, look, you know, that's what we've been going through for the past two years. It's all look over here and look over here. That could also be that symbolizes, just saying. So they're not, and we know, we know we got a Legion on our hands right now. That is the realization we have made is that we're up against the Legion right now of people that are like tentacles tied to what we think might be the same entity. We don't know, we just kind of know what we know and we just go with what we do know and that might, you know, but at the end of the day, in order to spiritually protect yourself, you got to trust your gut because God is talking to you. That's your source creator. That's the person that, that's the same source. Trust your gut. Your friend didn't create you. Your partner didn't create you. Your, the YouTuber you watched didn't create you. God did. So listen to that gut. And if you're confused about a situation, if you're feeling that discombobulation, step back. Step back from it and let it settle so that you know exactly how you feel. Again, no one's got to point things out to you. You are, I believe that every person watching right now is smart enough and is intuitive enough to be able to know exactly who and what we're talking about, all right? And if you don't know yet, maybe you haven't run across it yet and that's great, awesome video. You know, and just key, and understand that this is not, this battle is not over. And so you have to, if you were on a battlefield right now, you would not just be standing around eating popcorn, just willy-nilly, you would be guard, you would have your armor on. And we were talking about that, the armor putting, armoring up, right? Putting that light around you. Cause it is coming to the physical plane. This, this battle that has been going on all over the galaxy is now coming to the physical plane. I know the Cassiopeans that Todd follows, the off-worlders that have a forum, they talk about how they've been like excitedly watching Earth. It's strange. The Palladians, the woman who channels the Palladians, they have to, they've been excitingly watching Earth. And I'm gonna say what they said in one of their channelings, I'm gonna have to bleep myself and put the words up guys and you'll know why. We have, well, they have, the Palladians told the channeler, they have, that's how big this is. And they're watching us. Yeah, and whenever you look at it from like a higher perspective standpoint, it's affecting every planet in the universe. Like Mars getting liberated, the moon getting liberated too. There's a lot of liberation energy that's around us. And Earth is, Earth is really dense. So just remember that too, when you volunteered to drop in here, you volunteered for a really big task. So you're not doing anything wrong. And I noticed another pattern in my readings in my sessions was the cosmic womb. This feeling of like, I don't know which direction to go, hanged man in the tarot. It's a perfect example of it too. Right before you're about to jump through your birth portal, I know this all sounds cheesy. When I talk about the wombs and the portals and stuff, but it's a really good way to vision it. Think about being in a womb. What do you do when you're in a womb? Well, you take care of yourself. You trust yourself. You allow the energy to come into you instead of external and trying to, and that doesn't mean don't do anything, but what can you do that brings you joy? Or what can you do to help?  Like there's so many things you could do that resonate within your hearts and are within you. But I always say when you're in that cosmic womb, what can you do to take care of yourself? One, but to know that right before you jump through your portal, you always like want to jump away from it or like, and you don't know you're blindfolded until you completely go through, but you guys are blindfolded for a reason right now. It is not that you are doing anything wrong. I do just want to affirm that. We've gotten that a lot that we're all, that all of us, in other words, are cloaked. Like we don't have to, and that's for our best. That's for our safety and also for our own mental clarity to slowly start to reveal things that we're breaking a lot of patterns and a lot of stuff we now know our history is completely wrong. We know the Earth is not what we thought it was. I mean, that's a lot to take in. That's a lot of thought patterning to have to change. And so you have to remember that God is loving. The Pleidians are loving. They're not going to just be like, surprise in a bad way, right? It's going to be like surprise in a good way, right? So, trick ya. No, they're not going to do that, you know?  And I hate to be like, I kind of hate to be this way a little snarky, but like half of the human population doesn't even realize what's going on right now. I don't understand that. We are carrying the weight of the world on our shoulders right now because we are the ones that are aware and are fully activated in this battle. And so we have each other, though. You guys all have each other. We have each other because we are on the same team. Now, unfortunately, our side of the battle has been infiltrated, but we are aware of that. Some of us are aware of that. And the Galactics are aware of that. God is aware of that. And we will end up using this infiltration to our betterment because that's what God does because that's how clever and powerful the light is. Is it snarky? It's like, don't worry about it. I got it. I got through this. Also, thing two is like, do you really think God is surprised that it even happened? No. Number one, he allowed it. Number two, he already knew it was going to happen. And if you're going through what I'm going through, if you're one of the people, just know that, yeah, God is allowing this to happen for a reason and you got this. Sometimes you don't know how strong you are until you have to be strong. That's one thing my yoga practice taught me for years. I've said this before, my teacher, you know, when I strung a yoga, we put our legs behind our head, we catch our ankles, we need hyper mobility, hyper, but that's, but my teacher never talks about getting more flexible ever. He always tells you to get stronger. So you're in a handstand, holding a handstand, he's telling you to get stronger and you're sitting at the ground like, how much stronger do I have to get? But it's always get stronger, get stronger, get stronger. There's always that sense of getting stronger. And the more you've, and I said this at the end of my last catch-up video that I had said a while ago that when lockdown first started, I had this intense desire to start like strength training to do other things besides ashtanga. So I started really working on my strength. And I was under the impression that it was because I was preparing my body to go through the transference of the density of the earth. And that's what I was doing. That's why I was getting this message to do this. And for almost two years now, I've been doing strength training on top of my yoga. But through this battle, I realized it was kind of like, I was like, that strength training was for this. That strength training is what got your body through this battle you're facing with this entity. That's why I told you to do that because I knew this was coming. And so, again, that's following your intuition. In my intuition, I was being told to do this. It was nagging at me, do this, do this, do this. I didn't know why I thought it was just because we're gonna be, you know, I was all hippie-dippy about it. I was like, we're gonna go into fourth density and fifth dimension. No, no, it wasn't for the light and love. It was for the darkness that I had to get strong. You know, so, yeah, this is to swallow that gut. But ladies, I know we only have like 20 or so minutes left because Stephanie has to jump on the dark outpost. Do you wanna pull some cards? Sure. What do you guys think we should pull cards on? What do you feel like, okay, so what do you feel like the collective would wanna know? Like as we talk about either the dark magic or we talk about intuition or we talk about the womb, what do you guys feel like the collective would wanna know in this year? Let's go first question. Let's show what that, let's see what the spirit, the cosmos and the pleidians will tell us about the sprites to come. What, how what? Grace, you wish. I wish, right? Is that what they're saying? Well, ha ha, she's nosy, she is nosy. Guys, I think I've earned it. Come on, tell me, let me know in the secret.  I swear, I'll be like, buy cuts. Or maybe like, oh, you know what, just look what I just pulled. Stop, I think gotta go. I split my deck. Look at how red I am, I'm like. I split my deck at the full of the sun. Oh yeah, so that goes back to that, come to the edge, zero point energy. We were talking about what is your joy and that goes full circle to your intuition versus what feels good and what doesn't. So we're actually getting a full circle and the sun can be abundance in the tarot too, guys. So this could be what you've been waiting for as well. I get the justice card. Ooh. Two of cups or two of cups could be people coming together in my opinion sometimes because this is like a friendship card. People meeting each other. I also have the Wheel of Fortune and the Strength card. So I feel like this is saying that maybe many of us will meet each other in the actual flesh soon. That's what I feel too. I've been feeling that so significantly that there will actually be communities of like-minded people. And I don't know how that looks, guys. I don't think they'll even let us know how that looks. But all I know is that the financial strain or debt, what they were telling me pretty much was like, I don't need to worry about how it happens. It will happen, hold on to that knowing, but let go of how. Because that's when we have expectations and things like that that we need to release at this time. The next expectation is the peace of joy. So true. I just had something pop up over there. Hey, guys, how are you? Yep. Okay, so we've got a Queen of Wands energy which is following your intuition, your gut. This always reminds me of Brice, sorry. And a well-aspected Queen of Wands. She's very intuitive. But look at the sunflower right there. Whenever you're talking about that, that could be abundance. But also whenever we get yellow, it's also your power. Taking back your power, taking back your knowing, she's very passionate. And guys, we're about to literally level up. Look at him looking at the world. And this next step for him is always higher up. So that is your next step going through the portal. But the best part about it is you're walking away from the things that don't serve you. You're like, these swords don't serve me anymore. These humans don't serve me anymore. So this will be the five of swords. But it's walking away from the people, places and things that do not feel good. And guess what, guys? Mother Earth is assisting. We have so much assistance from the angelic realm, the galactic realm. We have Mother Earth as well. We're going to have a great, great healing come across. Like, I almost like, I'm sure, Steph, you probably had this vision too. Yours was purple. Well, mine was like a tsunami of energy too. So I'm sure we have mirroring visions of what's coming. But remember that this energy is coming from all aspects. But it's like Mother Earth wants to take her, like when I'm, Divine Feminine's coming together too. But sorry, I channel really fast. But Mother Earth wants to take her turn too. That's what I'm getting. And let's clarify, not the distorted Divine Feminine. This is a real lady. Real Divine Feminine, which is female and has state of female. As female and has state of female, yes. I'm just putting it literal. I'm very little. Actually, it was born in a female's body. Through the mother's womb. No distortion. I'm not bashing anybody who's open about it. It's when you're faking. Yeah. Distortion. Distortion and tricks, just like going back to the black magic, which... Yeah. That's that card for that entity. Yep. And that entity likes to wear red. We've now noticed so. Oh yeah, that's interesting too. Yeah, there's a bigger 111. There's a bigger, there's a bigger story to all of this. Of course, you know, Bryce is living this out personally, Stephanie being her friend and me now eventually like watching. But there's like bigger, bigger stuff to this. But guys, look at that. At the end of it, we walk through our portal. We get to celebrate together. This is literally like a celebration card too. So just know this is what's waiting for us.   I'm not one of those people.  I'm still gonna say that forever. But like, hold on to this knowing. Yeah. 111, as you say that. Ooh, we. Evan. Yep. Yep. I love it. I've gotten that too, because as you guys know, like as Taylor and Stephanie are friends of mine off camera as well. So we chit chat like friends. And I've said many times, like I don't know like how I'm gonna get through the hurt of being in this situation. And I just keep getting, it'll be healed. It'll be healed. There's gonna be healing. It'll be fine.  This is really real and it's happening. But there, there's gonna be clarity. We're gonna get clarity. I think we're gonna know everything. I think everything's gonna be, what is done in the dark, we brought to the light, right? I just got, the sun will come out tomorrow, but I'm not Stephanie, I can't sing. So you wanna sing? Guys, you guys have not heard me sing yet. I have not been able to sing lately because I've been so congested. My thoughts been really bad too. And sometimes I wonder if that's not, I mean, I know we film a lot, but I've been filming a lot for a long time. But I think that we're getting attacked. I think that's another place where we're getting attacked is through our work. There's also ascension too. I think you can get it with ascension sickness as well. I think there's a mixture. Yeah, we're getting it from all, like that's literally what I was singing is all angles.  And what that can mean is you have a family starting to try to connect with you too, that coming online too. I know it sounds, but that's what I was supposed to say. So I'm just gonna say it. I know that I had a weird dream. I've had three dreams within the last, I think month that a family member that has not talked to me, God knows how long that an immediate family member. And in my dream, I'm trying to talk to her and she just looks at me and won't talk back. It's really creepy. Like, what the heck does that mean? I don't think she's trying to connect with me.  Do you have like some stuff that happened with her whenever you started to learn truths? Was there anything like dramatic fallout with her or anything? That would be off camera. I'll tell you all about it. Steph, do you have multiple final cards for, I don't know Bryce, what do you wanna, what do you wanna feel like the collective wants to know in this here now moment? Let's look at the beginning. Let's, so we're moving into 2022, according to, we know that's not the year, but let's look at January. What do we overall feeling for January? Like for the, for like, so we're, are we checking like public collective or the collective of people that would be like, watching, watching you? What do you guys think? What do you wanna do? Public collective. Public collective, like baseline public collective. Let's see what the humans are feeling in January. I feel like I just got confusion. I'm sorry. I don't know if something's- I saw seven of cups in my head. I saw, yeah, so yeah, yeah, yeah, okay. So that means something might be breaking in January if they're confused. Okay. I'm still singing it guys. All I want for Christmas is my EBS. Let's, let's, we need to sing it. I'm telling you, I've been saying it. We got justice coming in pretty fast. Like this could be like that timeline of like it needs to kind of happen in this here now or else the timeline doesn't get perpetuated. I also think a lot of the humans, so we just got judgment with the three of swords. I don't want you guys to think you guys are gonna be through any more pain, but just be aware that people might choose to exit. People are going to be very confused. Everything's gonna be a lot because we're going into this like literally biblical time, right? Like this is, this is where we get our actual piece, not a thousand years of peace. Like that's what we're feeling. We're feeling like this is actual time. Yeah. And let's maybe, so let's, for those who missed that, we are now under the impression that our timeline is totally skewed and there's a possibility that the apocalypse already happened with the fall of Atlantis, which technically we're getting, might have only been like a thousand years ago. That's how messed up our timeline is. And so we've already had the a thousand years of peace with the Christ consciousness. And then if you read Revelation, the devil comes back for a short flip of time, which would have been the late 1700s, early 1800s. So now we are literally at the final battle. Is something as a theory that we've kind of been playing with, maybe we can draw a card and see if we can verify if that's correct. Yeah. Yeah. While she's doing that, I'm gonna show you what I got on the question you asked about January. So I get temperance and death. So I'm trying to word this correctly. It says the ends now for everyone. And this kind of goes along with what Taylor said. And I'm not trying to scare anybody either. But I've gotten downloads too that January might be an exit time for certain people. That aren't ready to ascend. And normally don't read the death card as death itself. This could also be death of the ego, death of religion, death of the old ways. All right. Death of money, death of all the systems that are out of alignment with the earth. Yeah. Anything that is out of alignment with the vibrational frequencies as we go up in frequencies. It's an ending. Yeah. And I did tap into the Atlantis 1,000 years ago and where we are in the Bible. Actually, I do believe that. I believe we're at the next step with the magicians. Whenever this is negatively aspected, I'm gonna take it that way. The magicians are the dark holds are the people who were in charge of us for a long time, bigger entities above. We're gonna walk away from everything they told us. And we're gonna have to start relearning our history. We're gonna get more. And so I just saw physical books. So those might be under the V. We might actually have access to those and start collecting all of our information. I do feel like like intuitively too, but also knowingly, Bryce had collected a lot of information about that stuff for a reason. And I feel like there will be a time when the world is going to really have to know what happened. But also in our acotias, we know like if you really tap in, like you know that they've been lying, right? Like it's a known fact that they, I was watching someone tour the Kennedy facility and they were all wearing these, which I'm all about free will. If you feel like wearing it, wear it, but don't make me wear it. Also, I don't think you should, but that's just my opinion. But at the end of the day, they were all wearing these and they were looking at this spaceship. And they were like, yeah, this has been 24 times out to space. And I looked at it, I go, that's a prop. It hasn't even been to space. It hasn't even touched space. This isn't true. This isn't what's happening. And it's just like the crumbling of like, what did they tell us that's real? At this point, we're gonna have to relearn and walk away from all things they told us about money, about ourselves, about our histories. Like, I know it's eight cups, it's gonna go on forever.  Like, there is like a firmament over our, whatever our planet is, isn't there? Yeah. It's like, I see it as being like a snow globe. Well, and also there's like rumors. I don't know this guys, I haven't tapped into this, but sometimes when I hear these little things, some of them resonate, some of them I need to tap into, but like, what if the challenger hit the filament and that's what happened? I don't know. I don't know, guys. I heard that that was a planned thing, by the way. I don't know either. I mean, I literally, I said on the video this morning, I did that if you had told me five years ago that I would be questioning what our planet looked like and what the solar system looked like, I would have laughed at you and told you you were crazy. But now at this point, I have no clue. And I'm comfortable with not having any clue what it is that we're standing on at this point. You said this before, and I actually think you should reiterate this to everyone too. Like, in my spiritual lessons, I'm sure Steph will agree with this too. My spiritual lessons, the best thing I've ever learned is that I know nothing. I can tap into things, but I literally know nothing. And I know you said that about your teaching yoga to people too. There's a huge surrender and realizing I have to question everything. And that's discernment too, question and call. That we say the easiest students to teach are the beginner students and the advanced students because both the beginner and the advanced student know that they know nothing. It's the intermediate students that think that they know everything. Another way to look at it too is this is the way I look at not knowing anything. I find learning is so much more fun now. Yeah, the information. It's a lot more sense. I mean, history's never made sense to me. I couldn't remember. Yeah, and like, I'll be honest, like I sucked at school, it never made any sense to me. I used to hate researching things, and now I love it. When I saw that picture of North America, I'll see if I can find an edit. It's the video here of North America possibly being like Egypt, Israel, Assyria, like all these middle, what we think of the Middle East actually being in the Americas, like our geography is so off. When I first saw that, I was like, yeah, that makes sense. Like it's just something inside of me, like it makes a lot more sense than maybe the cards. I already pulled cards on that. You have? What did they say? That's correct. I got that. That's very much correct. So we don't even know, like, I mean, think about that. I was even saying like, ah, so Yashua's tomb, we think is in the Middle East, but it's not. It's here. Well, we did some research, me and one of my buddies, she's great, but me and one of my buddies, we did some research on the Holy Grail too, and we found nine cups had been distorted and had little people's stuff inside the cups, and they were drinking out of them to try and distort the divine. It was, and I know that's disgusting, guys, I'm. Well, we have to face that. We have to face that because we picked up on that with these, and this Legion as well, that they're still doing these. Yeah, we think that in order to have their powers still available because they're weak guys, like there's a reason there's full blown attacks right now is because they're weak. We think for them to still have their powers that they're practicing still. Yeah. And that's disgusting and it should not be allowed, and I declare that should not be allowed. Me too, it's very important to know that these entities were planning this for quite some time now we found out. Yeah, at least a year. I've now was stalked for how long, I can't remember now, how long was I stalked for? Year to eight months is what it looked like, but you studied. I also feel like you'd always been kind of like watched though, like energetically in a protected way too, but yeah, you might have had a target on you. Yeah, I know, and I accept that, but like I said, I feel very protected. So whatever happened was allowed to happen, I get that now for my own good, and I'm really actually like right now, even though I'm still very upset about the friendship situation and that human emotion, I still understand that this probably I'm gonna look back and be like that was one of the most, that was probably one of the best things that ever happened to me, because it forced me. And that's one thing one of my friends said to me, who did a healing session with me as well, she said at the end of the day, what's gonna piss this entity off the most, if you think it. Yeah, you think it.   Well, that's taking back your power too. And I know, and I've gotten so many awesome emails, people I know you guys have said, oh my God, that I look better, I look more alive, I look healthier, and I feel it. So I think that that is definitely, I think that was one of the Pleiadians too. Like I think I definitely survived and learned from it. And I'm not, I mean, I'll keep learning, but I'll keep learning, and it's not over, it's not over. You know, as we said, this thing is gonna still continue to use my chart. Like it's not gonna stop using my chart, but I'm still, but I'm not gonna allow it to put me in the position it did. And I know my friend said after the Pleiadians took the track out of my neck, I had to hook some of my armpits because I can't get trackers in me now, so they're trying to hook me, but she took those out as well. So, so bring it. We are the same. Well, that's what I was literally about to say is we have the tower and we have the strength. Bada bing, bada boom. Don't be afraid of that tower. We need, we need that tower. I love the tower cards. We need that tower. It's divine. That's why I like the towers because whenever you see the lightning in the traditional tarot, it's always divine intervention. We have a huge divine cosmic plan. I mean, Steph even got the Wheel of Fortune, which is also what we've been seeing too. Oh, legacy. I've been getting this a lot. I wonder if everything is very temporary. Feels awful right now, but if this comes up, these girls can tell. This comes up with every spread I do with this subject. This, they're, they've lost out on something. It feels terrible. It's scary, but it's temporary. Fives are temporary. And there we have that infinity symbol right there. Yeah. That you are talking about. Oh, oh, I like that. I'm going to do our, our dance. Like, you know, you have a car lot.  Like, you know what? The devil doesn't know the sense of humor. Like we always talk about. So we got to have fun. Yeah. And that's energy too. Like that's alchemizing. We talked about this last time that laughing is literally the alchemy, like our trick of alchemization. So let's take what they've done to us, be like, have gratitude, right? Exactly what Brice said. Oh my gosh. And look at how strong you are.  Nothing can, you know what's, what's coming. You know what I'm saying? Yeah. Done. What did we say, what did one of our friends say on Monday night? It is done. It is done. It's done. Doesn't mean the devil's gonna just bow out and say, okay, bye.  And I really think- Laugh in his face. I think they've only got a few weeks left. If I had to guess, if I had to guess, I don't know for sure, but if I had to guess, like maybe a few weeks before. And when the tower falls, I think we're all gonna feel it. It's not, it's probably gonna be a little bit chaotic for, especially for the normal humans. Cause they're gonna be like, what? What do you mean? Adjun, can you imagine? No, but yes. Even some truth or communities, I mean, a lot of them don't know the mystical portion of this whole thing, the spiritual portion. They're very stuck in the literal facts. When it's, no, this whole thing is spiritual. Yeah. Yeah, it's all spiritual because that is the truth of who we really are. It's spiritual. It's such a good point too, because it's like collecting all aspects of all information now. Like you can't just keep your, you've got to look, you've got to feel, feel, go full circle back to what we're talking about. You've got to feel through it at this point. But it makes you nauseous as somebody gives you that awful feeling. Like, okay, nope. Trust yourself. Trust it. No matter what the channel says about that person, no matter what somebody else says about that person, you trust you. God talks to you too. All right. Let's end. What is that Bible verse from Esther, the book of Esther that's so poignant for this time? Perhaps you were born for a time such as this. Boom. Excuse me. We are the storm. Sorry, I got sassy there for a second. You did. It was great. Blind feminine right there. So guys, before we head out, I just want to let you guys know that I've talked to Stephanie and Taylor and I know we talked about last time doing some call-in shows the new year. So what I'm gonna do, I think the easiest thing to do because I am not technologically savvy is if I actually look at like StreamYard, which is another service that I can use where people, I can actually drop links into the chat so people can actually come into the chat to ask a question if we wanna do some episodes where you guys can ask Stephanie and Taylor some questions, look at some card stuff, all that kind of stuff. So we're gonna be looking at doing some more of those shows in the new year. Let me know your thoughts about that in the, I was about to say in the opinion section below in the comment section below, which kind of is the opinion section below. Let me know what you guys think about that. And I will definitely be putting both Taylor and Stephanie's channel links down in the description box below. So please go and give them a subscribe. Again, I think for all of us, numbers don't matter to us. We're not, this isn't a popularity contest. It's not a competition. We're all literally on the same team and we are all literally walking each other home. The only thing about the subscription numbers is it does catch the algorithms, which will push out their videos more for more people to actually run across and see. And that's super important in this battle that we get all the information out we can to those who need to see it. So we love all you guys very, very much. We hope that you all have a very merry, merry Christmas.   So go have fun and yeah, we'll arrange something next week for you guys. So merry Christmas guys. We will all talk to you soon. Bye. Bye.


So the afternoon, so last time we have been started to talk about rings. So we introduced rings and talked about subrings and units and so on. So it was all rather elementary. So now I will just proceed. So in particular, we want to start by introducing division rings and fields. So the first is a definition of a ring with the unit ring. So ring R with one is called a division ring. Well, if every non-zero element is invertible. So if R star is equal to R without zero. So R star, I recall, was the multiplicative group of units. So the elements which are invertible for the multiplication. So every non-zero element has an inverse. So we will in a moment not really deal with division rings. So we are more interested in the commutative division rings which are called fields. Commutative division ring is a field. So the last third or so of this course will be devoted to fields in particular to field extensions. So that means you have one field sitting, a smaller field sitting inside a bigger one. So I will write explicitly once again the actions for a field. So that we have them once together. So that means a set R together with two binary operations. So which are addition from R times R to R. And multiplication from R times R to R is called a field. Well, so two binary operations and two distinguished elements. So zero, which is different from one. The two elements zero and one which are different from each other is called a field. And first the addition, so R together with a plus is a commutative ring with neutral element zero. A compensative group, this was part of the definition of a ring. So second one is that if I take R without zero with a multiplication, this is a commutative group with neutral element one. And secondly, these two structures have to be somehow compatible. This is by the distributive law, so which just says that if I take A times B plus C, this is equal to AB plus AC for all ABC and R. So this is the definition of the field. As I said later, we will study this a little bit more. For now, so you know a few fields. No, we have Q, so the rational numbers, real numbers, and the complex numbers are fields. I want to also show that there's some finite fields. And for this I show that every finite integral domain is a field. So composition, every finite integral domain is a field. So recall that an integral domain is a commutative ring with one which has no zero devices. So there are no two non-zero elements whose product is zero. And we want to show that such a thing is a field. So that in such a thing, every non-zero element is invertible for the multiplication. So what we have to show, as it is already a commutative ring, we have to show that it is that the units are without zero. So in other words, we have to take a non-zero element in I to find its inverse. So we have to find its inverse element. Well, so we look at this distributive law. So the distributive law says, which we have in any ring, says that A, so now for this given A times E plus C is equal to AB plus AC. So in other words, that means if we take the map of multiple line by A from R to R, which sends an element B to A times B, this map is a group homomorphism for the addition on R from R plus to R plus. Because it says precisely that if I apply this map to B plus C, it's the same as if I first apply it to B and then to C and take the sum in R. Okay, so it's a group homomorphism and we know that R has no zero divisors. So in particular, there is no C in R without zero such that A times C is equal to zero. Obviously, A times zero is equal to zero. In other words, if I take this group homomorphism, then its kernel consists only of zero. If you have a group homomorphism whose kernel is just zero, it means it's injected. Now, we use the property that this is a finite integral domain. If you have a finite set and you have a map from this finite set to itself, which is injective, then it must also be surjective because you have as many elements on both sides, sometimes called the pigeonhole principle. So as R is finite, if the map is injective from R to itself, it must be bijective. In particular, if I take the element one in R, this must lie in the image. So let an element in R with A times B is equal to one. Well, so this exists because the map is bijective, but it gets surjective, so it means B is equal to A to the minus one. So we have found an inverse for our element for our non-zero element, and so it means that every non-zero element is a unit. Okay, so this gives us, we already know some finite integral domains because we had seen that if we take Zp, where p is a prime number, we know that this is an integral domain, and so, and thus it is a field, and it has with p elements. So for every prime number p, we have this field with p elements, and later we will see that for every prime power p to the n, there is a field with p to the n elements, and we will soon introduce the notion of isomorphism and homomorphism of rings and fields, and then we will find that for every prime power p to the n, there is a unique field with p to the n elements up to isomorphism. But as we will see later. Okay, so now we want to come to ring homomorphisms. So it's the usual thing whenever you introduce sets with a structure, you're interested in homomorphisms between them, which means maps between such sets which are compatible with the structure. Now for a group, the structure was the multiplication, and so the homomorphism must be a map between the groups which is compatible with the multiplication. And for a field, for a ring, you have two operations, addition and multiplication, so it must be compatible with both. So we talk about ring homomorphisms. So definition A and B be rings. So a map from A to B is called ring homomorphism. If it's compatible with addition and multiplication, so it sends the sum to the sum and partial product. So if for all A and B in A, we have phi of A plus B is equal to phi of A plus phi of B. Here's the sum in A, here's the sum in B. And phi of A times B is equal to phi of A times phi of B. That seems the obvious definition to make. And then we have the same words that we use for homomorphisms of groups. So the image of phi is phi of A. So that's the image as a map. And the kernel of phi is the kernel of phi, which is defined to be the inverse image of the zero element of B. So if you have a ring homomorphism, it's in particular a homomorphism of commutative groups if you just take the addition on both sides. And then the kernel is the same as the kernel was for this group homomorphism between the additive groups. So and, say, a bijective ring homomorphism, as usual, is a ring isomorphism. And if you have an isomorphism from a ring to itself, it's called automorphism. Phi from A to A is an automorphism. OK, I mean, this is a bit small, but anyway, it's the opposite. So now, as I just said, we have just defined the kernel to be the kernel of the homomorphism of the additive groups. So a ring homomorphism from phi from A to B is in particular a group homomorphism between the additive groups from A with the addition to B with the addition. And by definition, the kernel is the same as the kernel of this group homomorphism. Thus, phi is injective if and only if the kernel of phi is equal to 0, because that we know for the group homomorphisms of the additive group. And then, as usual, I don't even write it down. If you have a ring isomorphism, then it's straightforward to see that the inverse map is also a ring isomorphism. And the composition of ring homomorphism is a ring homomorphism. And last time, I introduced the notion of a suffering. So we also have that if phi from A to B is a ring homomorphism, then its image is a suffering of B. This is essentially directly from definition. It just means it's a subgroup of the additive group and the product of any two elements also lies in it. That was the definition of a suffering. And this is straightforward to check. If you want, you can check it as an exercise. So if you remember, when we were talking about groups, we had that the image of a group homomorphism is a subgroup, and the kernel of a group homomorphism is also a subgroup, but it's something better, namely a normal subgroup. And something similar is true here. If you'd have the kernel of a ring homomorphism, it's also a subgroup, but it's something better, namely an ideal. And so we want to introduce that now. Definition. So a subset in R is called an ideal if following holds. First, it's a subgroup for the additive group. So the sum of any two elements in I lies in I, and also the negative for any element which won't just the difference of any two elements in I lies in I. And the second statement is something more than for a subgroup, namely for all, say, x in I, and all A at x times A is in I, and A times x is in I. So to be a subring of R, we would have to know this for all x and A in I, we need xA is in I. So this is a weaker condition. So in particular, an ideal is a subring, but it's something that, so for the moment, so typical, so example, so same. And if the set 0 is an ideal, and if I take the whole of R, this is an ideal in R, this is basically obvious from the definitions. Let's now see what I just said before that I didn't write it. I just said it that the kernel of a ring homomorphism is an ideal, so let me state that lemma that phi from A to B, a ring homomorphism, maybe I should say that so here, as I'm not assuming that the ring is commutative at this stage, these two conditions are two different conditions, and I require them both. So that's for me an ideal. So in some books, you call that two-sided ideals, because multiplying on both sides, you still are in I. And then you also have left ideals and right ideals. But we will not be concerned with left ideals and right ideals, we are only concerned with two-sided ideals like this. Actually soon, we will restrict our attention to commutative rings where it makes no difference anyway. So let phi from A to B be a ring homomorphism, then the kernel of phi is an ideal in A. More generally, if I is an ideal in B, then its inverse image is an ideal in A. So first, you should be clear that this first statement follows from the second, because I've just told you, and it's anyway obvious, that 0 is an ideal in B. So I just applied to this, I get that the kernel, which the inverse image of 0 is an ideal in A. So now, I have only to show the second state. So let's take our ideal in B, and we just have to check the definition. So as i is a subgroup of the additive group of B, its inverse image is a subgroup of the additive group of A, by what we have learned when we did groups. So we only have to show this second condition. So let x be an element in phi to the minus 1. I and let A be an element in A. So we have to show that Ax and xA are in I. Then we have shown that it's an ideal. So if we take phi, this kind of, if we take phi of Ax, as it's a ring home of phi of A times phi of Ax, this is phi of Ax is phi maps x to an element in I. That's what's written here. So this is an element in I. And I multiplied with any element. So phi of A is some element in A. But as this is an ideal, it follows that this lies in I. And similarly, phi of xA element I. So it's basically obvious. And so this was this statement. So the inverse image of an ideal. Yeah, yeah. Sorry, sorry, sorry. That's what I obviously meant. I should also have written that. So here it's OK. And so Ax in phi to the minus 1 of I and xA in phi to the minus 1 of I. OK, thank you. So I want to say slightly more precise description to compare the ideals. If you have a subjective ring homomorphism, I want to compare the ideals. Say in A and B, if the ring homomorphism goes from A to B. It's again rather a simple thing, but it's useful to know sometimes. Lemma, or let phi from A to B be a subjective ring homomorphism, which sends an ideal in B to its inverse image is a projection from the ideals in B. The ideals in A which contain the kernel of phi. So we can describe all the ideals in B in terms of some of the ideals in A, namely those which contain the kernel. This is actually quite simple. We can write down the inverse map, namely just taking the image of an ideal under phi. So we define the inverse map. So this is the other way around. So I write again the idea from the ideals in A containing kernel phi to the ideals in A in B. Well, we want to use phi to do it. So the most obvious way we just take the image, so we take an ideal J in A which contains the kernel of phi is mapped to its image under phi. So I claim that this is the inverse map to this. So we know that this map sends an ideal in B to an ideal in A. And obviously, an ideal in A containing the kernel. Because this ideal contains the element 0, and its inverse element contains the inverse image of 0, which is the kernel. So the map does this. And now we want to write down the inverse map. So here's written down the inverse map. And now we want to see their inverse to each other. The first thing we might want to see is, however, that it sends that it actually does what is claimed here. That it sends an ideal in A, which contains the kernel to the ideal in ideal in B. Actually, it sends OK. So let's J be an ideal in A. I don't even need to have that contains the kernel of phi. Then we want to see that its image is an ideal in B using that the map is subjected. So to see phi of J is an ideal in B. Well, that's quite simple. Obviously, we know that phi of J is a subgroup of B with addition. Because phi is a group homomorphism for the active group, and J is a subgroup of A. So that's fine. So we have to only see that whenever I multiply with any element of B, I stay there. So let E in B and Y element in B of J. Then I can write, as the map is subjective, I can write B equal to phi of A, where A is in A, and Y equal to phi of X, where X is an element in J. Well, and then obviously, if I take, say, EY, this is equal to phi of A, phi of X, which is the same as phi of AX. And so it is in phi of J. And similarly, Y is an element in phi of J. So we see that if we have a surjective ring homomorphism, the image of an ideal is an ideal. So in particular, it holds under our X assumption that this ideal was supposed to contain the kernel of it. So now let J be an ideal in A, which is what we want to do. First, let I be an ideal. So we want now to show that this map is the inverse map. So we have to show that both compositions of this map with this map in both directions are the identity. So now we take I, an ideal in A. Do I want this? Yeah. No, I'd be an ideal in B. Then if we take phi of phi of to the minus 1 of I, this is obviously equal to I. In fact, that's true for any subset of P, which is just the property of being, this means all, this is the set of all elements which map to elements in I. And then I apply phi to it so I get I. This is by definition without any assumptions on what I is, obviously. And conversely, let J be an ideal in A containing the kernel of phi. And then we have to then it's, again, the following is clear. If we take the phi to the minus 1 of phi of J, this will contain J. Again, this is true regardless of what J is. If I have any subset of A, then phi to the minus 1 of phi of it contains the set. Because again, just by definition of phi to the minus 1, these are all the elements in A which map to elements in the image of J, in other words. And so certainly any element of J lies inside the set. Because that's, again, just the definition. So we have to see the other inclusion. So which one do I want to? So let's set an element in phi to the minus 1 of phi of J. We have to show, so we want to show the other inclusion, then we are done. So we have to show that Z actually lies in J. Obviously, if it's the fact that it lies in the inverse image of phi of J, means that if I take phi of Z, this is an element in phi of J. That means there exists an element X in J such that phi of Z is equal to phi of X. That's what it means to lie in the image of J. So if I take phi of X minus of Z minus X, so phi of Z minus phi of X, we know that this is 0 because they're equal. So this is phi of Z minus X. So it means that Z minus X is an element in the kernel of phi, which we know is contained in J because that was our assumption. So thus, we can write Z, which was our element, which we wanted to show is in J. You can write this as X plus Z minus X. And this is an element in J. This is an element in J. So this is an element in J. So our element Z is in J. So we find that phi to the minus phi of J is equal to J, and which means that this map is inverse to this. And so they are bijective to each other. OK, so this is again a quite elementary thing. So today, I mostly do very simple things, but also history. I just start again slowly, and then we come to more. I'm more interested in things later, but not really today. So another thing that one has is we had defined the subgroup generated by some element. So we can also look at the ideal generated by some elements in a ring, which is in some sense easier. So the definition, let R be a ring, and say A1, 2, An, some elements in R. The ideal generated by them is just, so it's denoted like we had denoted the subgroup generated by it, but always be clear from the context which one we mean. In this case, it's just a set of all linear combinations of these elements with coefficients in R. So this is the set of all A1, R1, plus An, Rn, R. Yeah, so I was not. So I want this to be a commutative ring, because otherwise this will not work. And we have our elements A1, 2, An, in R. So with this, we look at all these linear combinations where the Ri are elements in R. So we just look at all these combinations. And so it's easy to see. It's very trivial. This is an ideal in R. Of the simplest cases, obviously, that this ideal is generated by only one element. So for A in R, the set A, the ideal generated by this one element, which I could also just write as A times R, which is a set of all AR with R in R, it will be called the principal ideal generated by A. So for instance, in Z, we have N, which is just equal to NZ, which are all elements divisible by Z. We will see later that in Z, all ideals are principal ideals. So every ideal can be written as an ideal generated by just one element. And the ring with this property will be called the principal ideal ring. But as I said, we'll see that later. So just one. So for security, say let R be an integral domain. So we find that the principal ideal generated by an element, by two different elements, will be the same if they only differ by multiplication by unit. So that A and B, the elements of R, then the ideal generated by A is equal to the ideal generated by B. If and only if there is a unit, a invertible element for the multiplication, U in R is equal to U times A. So it's easy to decide when two principal ideals are equal. That's again basically obvious. So if A is equal to B, then it means in particular that B lies in the ideal generated by A. So it's a multiple of A. Then B is equal to U A for some U in R. The only thing we have to see is that this U is a unit. Well, we also have that A lies in the ideal generated by B. So we also have that A is equal to W times B for some B for some W in R. So if we put this in here, we have that B is equal to W, so to U times A. And A is equal to W times B. But as we are in an integral domain, we can cancel the factor B. So it follows that 1 is equal to U times W.  So this shows this direction. This direction we have to, is in some sense, obvious if they are related in this way, we want to say that they generate the same element. Conversely, if B is equal to U A with U a unit, well, first B lies in the ideal generated by A, because it is A multiplied by something. So the ideal generated by B consists of all elements of R which you obtain by multiplying B by something. But these are then also, that means I can also multiply the, so this is all W times B, but it's all the same as all W U times B. So it follows that the ideal generated by B is contained in the ideal generated by A. And now, obviously, if B is equal to U A, then A is equal to U to the minus 1 B. And so we can change the role of A and B. So also, equal to U to the minus 1 B plus A is contained. So first, A is an element in B, and thus, A is contained in the ideal generated by B. OK, this was another simple remark. So then, we want to see that the ideals, so in a field, they're essentially no ideals. We know that in any ring, the set consisting of the element 0 and the whole of the ring are ideals. And in a field, the claim is these are the only ones. And this also will lead to the fact that any non-zero homomorphism which starts from a field has to be injective. So let's state this in Mark. So let I be an ideal in a ring. So if this ideal contains a unit, then it is the whole of R. Second statement, which is an easy consequence, is that the only ideals in a field, or k, are 0 and k. And the third, which is a consequence of that, is that, so let k be a field, and assume we have a homomorphism from k to any ring, a homomorphism to a ring. Then either phi is injective or phi is the 0 map. So by this, I mean that phi of x is equal to 0 for all x and k. Or phi is injective. So this is all very simple, actually. So let's do the first one. So assume we have such an element which lies in i and is a unit. So let A be an element of i, which is also a unit. So i is an ideal. So the inverse element of A lies in R. So if I take A to the minus 1 times A, this will lie in i. Because the product of an element in i with an element in R is equal to 1. We find that the element 1 lies in i. And so thus, if I take any element x in R, then I can write x equal to x times 1. And so this is now a product of an element in R with an element in i. And so this is an element in i. So this is the first one. Second one is a direct consequence. We know, I mean, we use that the units in a field are precisely all the elements in the field which are not 0. This is the definition of a field. And so if I have an ideal which does not only consist of the element 0, it contains a unit. And therefore, it is equal to the rule of k. So this follows directly from 1. And the third one is a direct consequence of 2. Because the kernel of phi is an ideal in k. And thus, either the kernel of phi is equal to 0, which means phi is injective, or the kernel of phi is equal to the whole of k. It's the only other ideal, which means phi is 0. So when we were discussing normal subgroups, one thing that one could do with normal subgroups was that you could take the quotient group by the normal subgroup. So the set of cosets by this normal subgroup was naturally a group. And so as I said, the ideals are somehow analogous to the normal subgroups in a group. So we also want to take the quotient of a ring by an ideal and see that that is also a ring. So let me do this. Let R in ring and I in ideal. So we know in part of the definition of a ring is that the additive group of the ring is a commutative group. So R comma plus is commutative. So I, which is a subgroup of R comma plus, is a normal subgroup. So we do have the quotient group for the addition. So thus we have the quotient group, R mod I, which consists of all the cosets with respect to I. So this set of all X plus I with X and R, where X plus I is just the other word for the equivalence class. So also the equivalence class of X for the equivalence relation as we had before for the groups, that X is equivalent to Y if and only if the difference is an element in I. Then the equivalence classes will be precisely these. And the set of equivalence classes is this quotient set. And this is a group. So this is, as I said, is a group with X plus I plus Y plus I in defined to be X plus Y plus I. So in future, however, to simplify rotation, I will always write this just as equivalence class like that. And now we want to show that this thing is also a ring. So we're going to call it the theorem. That's the Godfrey composition. So we have the same situation for R as a ring. And I and R is an ideal. So R mod I with the operations plus Y, this is what we have just written, equal to X plus Y. And X times Y is a ring. So the zero element will be R of 0. And the natural projection, we had the natural projection before, just a map which sends every element to its equivalence class. So pi from R to R mod I, which sends any element X in R to its equivalence class or its coset, is a ring homomorphism. So this is very similar to what we did for the groups. It's also not more difficult. We just have to. So essentially we also have already seen it. So we have seen that so that we get an additive group if we just take this sum. And we have to then therefore show that the product is well defined and makes this thing into a ring. And that this is a ring homomorphism. So this is a ring homomorphism. And obviously it is a surjective ring homomorphism. That the map is surjective is obvious because this map is obviously surjective. OK. Not so high. So by what we have just seen, we have that R mod I together with this addition that we had defined. So we were right to do this as before with X plus Y equal to X plus Y is a commutative ring, commutative group. And this map pi is a group homomorphism. So from R plus to R i plus is a group homomorphism. This we have already seen because this we already know from what we did about groups. OK. Maybe I should also say here that this thing has, this homomorphism has obviously a kernel. So we can, given any ideal, we can find a group homomorphism by taking this quotient so that the kernel is that ideal i. And obviously this is a group homomorphism with kernel i because the inverse image of 0 is the equivalence class of 0, which is i. So we want to show the product is very defined. So that means if we have two different representatives for this class of X and the class of Y, we still get the same thing for the product. So if the class of X is equal to the class of X prime and the class of Y is equal to the class of Y prime, then this means in other words that X minus X prime lies in i and Y minus Y prime lies in i. So then if I have XY minus X prime Y prime, I can kind of write this in a different way to show that this difference also lies in i. In the I write this as X minus X prime times Y minus, I hope it's correct. Anyway, you can check this, X prime times Y prime minus Y. So anyway, the things that I want is this one, this one, and these two are the same. So that's OK. And you see that this thing is an element in i and this thing is an element in i. So it follows that this whole product lies in i and this whole product lies in i. So the whole thing lies in i. That means the product of X, the class of product of XY is equal to that for X prime, Y prime. So the product is very defined. Only depends on the equivalence class. And then to see that, so the product is very defined. So to show now that this gives us a ring, we have to show associativity and distributivity. Associativity, the associative law and the distributive law. Well, this is completely trivial because it follows directly from that in R. So if I just do it for the associative law, so I want to say I take A times B times C. My definition, this is A times B, C. And this is the same as A, B, C. Now you can already see that we don't remember the brackets anymore, so it's obviously anyway A, B. OK, so it's just moving around these dotted brackets, so it's kind of clear. The associative law and the distributive law is as simple. And at pi as a ring homomorphism is also obvious. So if I take elements, so for x, y in R, I have pi of x, y. My definition is the class of x times y, which is class of x instead of y, which is pi of x times pi of y. So this is all. So anyway, so we find this quotient ring. And so if you have an idea, you can divide by the idea. You get a new ring, which works precisely in the obvious way. OK, so we can, if we want to, in particular, so we have this ring homomorphism from R to R mod i. The kernel is i. So we can, therefore, describe the ideals that contain i, a rolling projection between the ideals of R mod i and the ideals in R containing R. As we've seen, the projection is given by taking the inverse image of the natural projection. OK, now finally, how much time? Finally, I want to show also the other thing we showed for groups so that we have the homomorphism theorem. And we can basically, if we have a homomorphism between rings, and then we can divide by any ideal, which is contained in the kernel. So again, it's not really so much of a theorem. But anyway, it's a universal property of the quotient. So let phi from a to b be a ring homomorphism. And do we take an ideal in A, which is contained in the kernel of phi? Then we can, one says then one can factor the map over the quotient by i or through the quotient by i. So then there's a unique homomorphism phi bar from a mod i to b. Such that you either write this as a commutative diagram, such that this diagram A phi b, you have the natural projection to a mod i. And here we have phi bar commutes. This just means if you go this way, same as if you go first down here and then there. So in other words, phi bar is equal to phi bar composed of i. And we can see furthermore, the kernel of phi bar is equal to the kernel of phi modulo i. So the quotient by this ideal. And the image of a mod i, the image under phi bar is the same as the image of phi. Well, this takes much longer to state this than actually to prove it. If you look at it, we have imposed this condition that phi bar should satisfy this property that I have composed it with phi, with phi, then I get phi. But this will determine the phi bar first. So the uniqueness of the homomorphism will be obvious. Namely, so if that phi equal to phi bar composed with pi, this fact means that if I take phi bar of any element x, so of any plus x of any element in a mod i, this has to be, by this formula, equal to phi of x. So if phi bar exists, it is unique, because it's written down the formula for it. And obviously, we'll use this definition. We only have to define to prove it's well defined. We have to show with this definition it's well defined. Well, and that's kind of obvious. So if the class of x is equal to the class y, then, as we have seen now a few times, x minus y is an element in i, which is contained in the kernel of phi. So it follows that phi of x minus y is equal to 0, which is equal to phi of x minus phi of y. So that's it follows. So it means that this phi bar is well defined. The image of the class depends, does not depend on the representative. You always get the same result. And it's immediately to check that this is a ring homomorphism. We've done similar things already a couple of times today. You just check that if you take the sum, it maps to the sum. If you take the product, it maps to the product. This follows immediately from the fact that phi is a ring homomorphism. And then it is clear we have to find phi of an equivalence class to be phi of the representative. So therefore, the image of phi bar is the same as the image of phi. Because every element in the image of phi is obtained. This way, we just take any representation. We take the class for any class. So if we have something which is in the image of phi, we just take the equivalence class of the thing that maps to it, and this will map the phi bar. So really by definition. Basically, also by definition, we have that the class of x is in the kernel phi bar, if and only if phi bar of is equal to 0. This is equal to phi of x. So that means if and only if phi of x is equal to 0 for A for 1 or equivalently for every representative of x, so this means if and only if. So if and only if x is in the kernel of phi, which is if and only if the class of x is in the kernel of phi, or to go up. Because we have just the equivalence class that just taken the class, or to go up. So maybe I can also finish this section by finishing with the homomorphism theorem, which is a straightforward equation. It says, like with the groups, that if we have a subjective ring homomorphism with kernel i, then this gives us an isomorphism of the quotient by i with what you are mapping. So let phi from A to B be a subjective ring homomorphism with kernel i. Then the map phi bar from A mod i to B is an isomorphism. So in particular, A mod i is isomorphic to B. And that's basically clear from the universe property. So by the universal property, we have that. So this phi bar is a, so this map was a subjective. So phi bar is a subjective ring homomorphism with kernel. What's the kernel? So the kernel of phi modulo i. So the 0 is i mod i. So the kernel is just a 0 element of A mod i. So that means it is also injective. So maybe I stop. So I have, as you see today, we did not really do anything interesting. Maybe I went a little bit fast because of it. I don't know. On the other hand, there was no argument which was not very easy. So just everything we did was kind of analogous to what we did for the case of groups. And the proofs are very simple, similar. So hope next time we will do a little bit more. We will talk about prime ideals and maximal ideals, which is creeper and principle ideal domains. So anyway, we'll see each other on Friday.