back. I also just want to encourage everyone to go to the TA sessions. I think it's really, you know, it's really good to take these sort of problem sets. Some of the problems are really just sort of reinforced or some of the discussion topics are to sort of reinforce the material lecture and somewhere to go a little bit further. And even if you haven't started the problem sets or you have lots of questions or you want to work on a previous problem set, you're very much encouraged to go to the TA sessions and the TA's are very happy to talk not only about the most recent problem sets but previous ones or answer any questions. Okay, so let me set up the screen chair. Okay, so yeah, so I want to sort of pick up from last week. So right, so last week or towards the end of last week, we introduced in the field of piatic numbers. So this was a suitable completion of the rational numbers with respect to the piatic absolute value. And right, so I guess we discussed proof of are we discussed Hensel's lemma that gives a criterion for when you can solve equations in the piatic numbers. And at least for p greater than two, we discussed the classification of quadratic forms over QP. And so let me let me just recap how that how that went. So so the main result was, well, so you can you can say that in terms of bit rings, for example, so as a classification of anisotropic quadratic forms over QP when P is greater than two. And the statement is that the the bit group, so let's say the bit group of QP is isomorphic to two copies of the bit group of FP. So in other words, an anisotropic quadratic form over QP is a sort of equivalent data as a pair of anisotropic forms over over FP. And so specifically, right, so where does how does this correspondence come about? Well, if you have an anisotropic quadratic form over QP, well, you can write it in diagonal form. And then you can assume that it's, so you can assume it's it looks like, say, you one through you are comma PV one through P vs where the UIs and the VJs are in ZP cross. So by sort of multiplying by even powers of P, you can always remove the diagonal entries of their piatic valuation as either zero for a periodic units or one, so it's P times a Patic unit. And so the class of this is going to go to the class of U one bar us, sorry, you you are bar, so you sort of reduce mod P, and then V one bar through VS bar.  And so this gives a correspondence between anisotropic quadratic forms over QP and pairs of anisotropic quadratic forms over FP for P greater than two. So let me just write that again. And yes, so this is what this is a special case with your move Springer. And so this was this was really proved using Hansel's line, you needed Hansel's line because somehow you're, you're taking square roots. And, well, it's easier to take square roots when, when in QP for, for, yeah, so thanks. So, right, so where did we use P, P is greater than two, well, we use P is greater than two. So the fact that this is even well defined, so it's not obvious that something like this is even well defined, first of all, because, you know, you can, there are many different ways you can write a quadratic form in, in diagonal form. And so it's not obvious that this kind of construction is even well defined, begin with. I guess, in fact, the way we really constructed it was more as a map from right to left, that if you have a quadratic form over FP, then you sort of lift it to a quadratic form over ZP, and then QP. And, and so that you could see was sort of well defined by really by using Hansel's sum up, because so, so for example, you're really using the structures of squares, right, so maybe I should write this so this really used the structure of squares in QP. So in particular, so we use the fact that any element, let's call it X, which is congruent to one mod P, is a square in QP, right, say in ZP. And so that's going to imply that the map from right to left, which is I guess maybe the most more natural way of actually defining the map is well defined. Right, so this is again, so this is for P greater than two. Right, so this, right, so this gave a pretty sort of concrete, but yeah, so the conclusion was that you have a pretty concrete sort of parameter parameterization of anisotropic quadratic forms over QP. And in particular, a corollary is that any anisotropic form over QP, again P greater than two, has dimension less than or equal to four. Right, because we know that any anisotropic quadratic form over FP has dimension at most two, any three dimensional form over over Q over FP is isotropic. So that was, oh yeah, so that was a little bit of a recap from Friday. And the first thing, yeah, the first thing I wanted to do today was to try to sort of revisit this, this type of statement. And in particular, I mean, this is really for QP for P greater than two. But in fact, there is, there is sort of, so this particular statement isn't quite going to work in in characteristic two or sorry for P equals two. For example, we haven't even defined the bit, the bit ring of F2, we've only considered bit rings in an odd characteristic or zero characteristic. But in fact, there is sort of a uniform, a somewhat uniform description of the bit ring of QP that works even when P is equal to two. Well, at least sort of modular filtration. And in fact, it doesn't just work for QP. It also works for any local, any local non or comedian fields. So I mean, the first goal today is so to describe the bit ring of QP in a more uniform way. Sorry, so yes, yes, thanks. So in the question, right, so yeah, so let me write it this way. So in other words, the un variant of QP is at most four. Well, actually, it's equal to four. It's equal to four because you do have an anisotropic quadratic form over over FB. And then you can lift that.  And so by this, you get, yeah. Okay. Right. Right. So and I guess another thing that you can say is that there are exactly 16 isomorphism classes of anisotropic quadratic forms over QP for P greater than two, because the bit group of FB has carnality four. I guess that was something right that we saw earlier. Okay. So first is also I want to say something about, you know, like the PI like numbers in general and right. So, yeah, so in fact, it's not it's not just going to work for for the fields of PI numbers, but it's going to work for any any local fields. So I want to make the following definition. So definition, a local field is a field E with an absolute value. So with a non-trivial absolute value, well, up to equivalence, right. So we said that two absolute values are the same or equivalent if they give you the same topology or one is a power of the other. Right. And such that E is locally compact for the induced topology. So as a metric space, it's it's locally compact. And so in particular, it's complete. Right. So for example, we've seen that that QP is an example of a local field. So I think this was on the this was on one of the problem sets. So if you take the the unit ball or the unit desk, so that's the piatic integers in QP, that's a compact space. And well, also the real numbers is a local field. Well, again, the real numbers are locally compact. And similarly for the complex numbers. Right. But so in fact, there's a classification of local fields. So essentially, there are three different types of local fields. So, right. So let E be a local field. So the first statement is that if E is Archimedean, so if you have a local field that comes with an absolute value, and you can ask if that's Archimedean or non Archimedean. So Archimedean, the non Archimedean meaning that you have the stronger absolute value of X plus Y is at most the maximum, the absolute value of X and the absolute value of Y and Archimedean, the condition that that is, well, that's not true. So if E is Archimedean, then E is isomorphic to the real numbers or the complex numbers. So that's sort of a first type of local field. Yes, isomorphic as a topological field or as a field with an absolute value with just to find up to it. Yeah, thanks. Okay, so if you have an Archimedean local field, then it's it's the real numbers or it's a complex number. So that's great. And then second is the case if E has positive characteristic. So if then E is isomorphic to the field of Laurent series over a finite field. So another way that you can cook up a local field is that you take a finite field and then consider all Laurent series over that field. And so the field of Laurent series over a finite field, well, it has a natural absolute value, which is the tiatic absolute value. Where instead of, well, I mean, right, so where you look at so the valuation of the absolute value of something is going to be in terms of how many powers of ti divided or what sort of the order of the zero or pole of this Laurent series at a t equals zero. And right, so that's going to give you that's going to give you an absolute value. And in fact, it's a local field. So for example, if you're in a situation like this in the unit disc, the elements of norm at boost one is given by fq double brackets t, the ring of formal power series over fq double brackets t. And that is given the tiatic topology. So that's given. But if you think about it just as a topological space, it's a it's a product of copies of fq. So as a topological space, the unit disc looks like a product of copies of fq, accountable product of copies of fq. And that's that's compact. Well, I guess it's a special case of taken off serum. It's a product of finite sets. But in fact, it's accountable products, it's really sort of maybe a more explicit case of taken off serum. Okay, so then this is this is a case of positive characteristic. And right, and then finally, there's the case of characteristic zero, but but non Archimedean. So if the characteristic of E is equal to zero, but E is non Archimedean, then E is a finite extension of QP of some QP. So so the periodic numbers are an example of the local field. But in fact, if you have any finite extension of the periodic numbers, any finite field extension. So for example, you could take something like QP, and at a square root of P, or at a P through unity or something, and you're going to get some sort of finite extension of QP. And then it's a basic theorem that you can always extend the absolute value to this finite extension. And it's going to become an example of a local field. So for example, I mean, there's essentially, there's going to be a unique way of extending the absolute values of the absolute value of the square root of P, for example, is going to be square root of the absolute value of P. And, and then it's going to write so it's going to be a local field. I'm sorry, so it's a question. Isn't there with respect to what you just said, just to, to see whether like this stems from a general theorem, isn't there a general result that says about this extension of the the absolute value that if I have an algebraic extension of a complete field, which has some absolute value, then I can uniquely extend it. I think it's like cruel extension theorem, right? Or not exactly sure who would. Well, that's true. That's correct. So if you, if you have a, if you have a complete field and even algebraic or finite extension, you can always is a unique way of extending the absolute value. Thanks. It's not unique. If it's not complete, but you can always do it. But if it's complete, then yeah, thanks. Yeah, thanks. Okay. So, right. So I guess I wanted to kind of, you know, sort of say this explicitly for a couple of reasons. One is that some of these results are going to really work uniformly for local fields. So not just for QP. And so it's good sort of stadium in this generality. But also, like if you, if you haven't seen QP before, then maybe this sort of also helps to sort of orient orient yourself because the idea is that if you look at, if you look at cases two and three, the idea is that QP, you should think of QP as sort of analogous to the field of formal, formal Laurent series over a finite field. And so in general questions involving, you know, various types of questions, you expect them to be sort of easier for formal Laurent series than for QP. So QP, I think of it as sort of maybe slightly more difficult version of formal Laurent series. And for example, it's more difficult because, well, if you think about piatic numbers, right, I mean, so you sort of informally a piatic number is something like a formal Laurent series in the variable P. But when you add piatic numbers, you have to carry. There's some sort of carrying involved. Whereas if you add formal Laurent series in the variable T, there's no carrying involved. So, so informally QP is like Fp Laurent series T, the field of formal Laurent series in a variable T, except the variable that quote unquote variable is P. And somehow when you add and multiply the rules for adding and multiplying is going to be a little bit more, going to be a little bit more complicated. I mean, in practice, it's probably best not to think of QP as literally sort of Laurent series in P is this really more of an analogy. I mean, really you build QP using this completion procedure. But there's just sort of an analogy. So what I wanted to do is to try to describe the bit ring of, so in general, there's sort of a description of the bit ring of any local field or at least a description up to sort of associated graded terms. And yeah, so I wanted to sort of explain that. So in general, there is a pretty explicit. But to explain this, we need to introduce one more ingredient, which is called the Hilbert symbol. So this is going to use an ingredient called the Hilbert symbol. So I guess my next goal is to sort of explain that. But maybe let me sort of state one of the main, so let me start by sort of stating like one of the main sort of outputs of this classification. So theorem, so let E be a local field. And so first of all, I should say that if E is the real numbers or the complex numbers, we already know we saw sort of really in the first lecture what quadratic forms look like. So let's say E is a non-archimedean local field. And let me assume E has characteristic greater than not two. So Q2 is okay. Q2 is characteristic zero. But I don't want to consider F2 Laurent series T because we haven't really talked about quadratic forms in characteristic two. So then what is always going to be true is that any quadratic form in at least five variables is isotropic. And there exists a unique anisotropic form of dimension four. So in particular, the univariant is going to be equal to four. Okay. So with this in mind, I want to try to say something about the bit ring of E. So goal is to say something of the field E. And so in general, if F is any field, we can consider the bit ring of F. So this is the ring of all isomorphism classes of quadratic forms, modulo the hyperbolic forms. And the bit ring of F has an ideal, has a natural ideal sitting inside it, which is the ideal of forms of even dimension. So I is additively generated by the forms, well, I'm going to call them the forms brackets one minus a, as a ranges over the elements of F cross. So just sort of for reasons of normalization, I'm going to put in a minus sign here. And this ideal I is additively generated by these two dimensional forms. Because if you have any two dimensional form, you can write it as sort of the difference of these things. Well, if you have any even dimensional form, you can write it as a sum with signs of these things. Okay. So, right, so you always have this ideal, this is sometimes called the fundamental ideal. And right, so moreover, what you can do is you can describe, you can describe, so here's a basic fact about this ideal, which is that there is an isomorphism, which is that i mod i squared is isomorphic to F cross modulo F cross square. So that, so if you take i modulo i squared, then it's isomorphic to the nonzero elements in F, modulo the squares in F. And this isomorphism is coming from the signed determinant or the or what I guess what I or the the discriminant. Right, so sorry, so there's a question. How do we know that over E we can always represent one? So we can't, I mean, I guess the statement is that if you have, right, so I mean, if you have some form brackets, you know, B comma C, then you're supposed to write that as a as a, right, so sorry, so this is this is saying that as a as an abelian group, it's generated by brackets one minus brackets a. So this is also in the bit ring, this is the same thing as, as one, one minus the class of brackets a. And so if you have, if you have some even form, which is like a comma B, or brackets a minus brackets B, you can write that as a difference of these times these types of elements. Yeah, so, so explicitly this this isomorphism sends the class of an even dimensional form, let's say V comma Q. Well, it's going to go to it's going to go to the determinant of V, but it's really going to be up to a sign. So it's going to be minus one to the dimension over two times the determinant of V. And the claim is that this induces an isomorphism between I mod I squared and the units mod, mod the square of the units. So, yeah, so the reason for the signs basically is that the is that you're working with a bit ring instead of the grown G at that ring. So, so, so the bit ring is you're taking all quadratic forms, but you're allowed to add on a hyperbolic form and a hyperbolic form is not going to change anything. So the hyperbolic form has determinant one, but it has dimension two. And sort of because of that normalization, because if you add on a hyperbolic form, you'll change things, you'll change the determinant by a sign, but you'll change the dimension by two. You define the sort of signed determinant or also called a discriminant. And it's well defined on the ideal of even dimensional forms in in in the bit ring of F. So, so this. So this is a map called the discriminant which goes from I to F cross mod F cross squared. And now I squared the square of this ideal I is generated by products of elements in I so it's generated by things of the form one brackets one minus A minus B, AB. And these classes have discriminant one. So it actually factors over a map which goes from I mod I squared to F cross mod F cross squared and then one one checks that that's an isomorphism. And so the map in the other direction is right to the map in the other direction which goes from F cross mod F cross squared into here is sending A to brackets one minus A. So this is something that's true for any field for any field. For any field this augmentation ideal modulo I squared is automatically isomorphic to F cross mod F cross squared. So, so let me just sort of recap that by saying the general facts. So for any field F, the idea is to consider the augmentation ideal I sitting inside the bit ring W of F and the iatic filtration. So and the sequence of powers of this ideal I so you have the bit ring of F that contains the ideal I and that contains the square of the ideal and so on and so forth so you have this you have the sort of descending filtration of the bit ring. And you always have, for any field F, you always have the formula that I mod I squared is isomorphic to F cross mod F cross squared. Well, first I should say that W of F mod I, well, I is the ideal of even dimensional forms so W of F mod I is Z mod 2. And I mod I squared is F cross mod F cross squared. So now I want to explain what this sort of gives you if you have a local field. So for a local field, in fact, you have the formula that this filtration is actually finite. So for a general field, this filtration might be infinite, but for a local field, what you always have is that I cubed is equal to 0. So for a local field, let's call it E, you always have that the cube of the of the augmentational ideal is 0. So this is, I guess, closely, I mean, this is sort of coming from the fact that five dimensional quadratic forms are automatically isotropic. So this is coming from the U invariant. The U invariant of E is at most, well, it's equal to 4. And in fact, and in fact, I squared mod I cubed is a Z mod 2. So for E a local field, when we have this finite iatic filtration on the VIT ring and the associated graded terms. So in general, if you have an ideal, I mean, if you have an ideal and some commutative ring, you can I mean, you can look at the the terms of I to the n mod I to the n plus one. Right. So you have W of W of E containing the ideal I containing the ideal I squared and then containing zero. So I squared here is equal to Z mod 2. The the the VIT ring of E module I is also equal to Z mod 2. And I mod I squared is equal to E cross mod E cross squared. So there's this description, if you have any local field. So there's not sorry, right. So sorry. So there's a question which is, do we already know that local fields have you invariant for so no, I have not proved that yet. So what we've proved is we've proved that for QP. So yeah, so thanks. So so just just to be clear about the logic here, I guess I'm not probably going to be able to prove everything in this lecture. But I just want to describe. So what I'm going to explain is going to be stuff that we've sort of entirely proved for for QP for P greater than two. And I want to sort of try to explain how this so we haven't proved it for Q2, but it's not actually too hard to prove it for Q2. So for Q2, yeah, right. So so for Q2, this is kind of right. Sorry, I think I should have experienced earlier. So for Q2, I think I will put this on the problem set. Yeah, so just just to back up a little bit. So yeah, sorry for that. So for Q2, I guess I want to sort of put this on the on the problem set. And so it's fun to actually work work all this out pretty explicitly. So can show directly that every five dimensional form is isotropic. And that any anisotropic four dimensional form is isomorphic to the following, it's brackets one minus five to minus 10. So so for QP for P greater than two, you can work this out sort of abstractly using Henselzama. But for Q2, you sort of have to get your hands a little bit dirty.  And the key thing that you need to do is you need to essentially look at congruences instead of looking at congruences mod P, you need to look at congruences mod eight. So if you want to prove something like every five dimensional form is isotropic, you need to look at congruences mod eight. And that's because any element of z two, which is congruent to one mod eight is a square. So in fact, the squares and z two cross are exactly the elements which are congruent to one mod eight. So z two cross squared is exactly those x and z two with x congruent to one mod eight. Yeah, so sorry, so thanks for asking. So I'm not going to try to give. So in this lecture and in tomorrow's lecture, I think I would not try to give these proofs in general for a local field. So it's actually right. So it's not too difficult to prove it if you have a local field, which is a finite extension of QP for P greater than two, more or less the same proofs over QP for P greater than two are going to work. But it is going to be a little bit tricky to prove these types of things completely directly. If you have if you have an arbitrary finite extension of Q two. And a lot of it is really subsumed under under local class field local class field theory and all this really sort of follows as a special case of somehow of local class field theory. But for Q two, I think it's a good idea to actually check directly that every five dimensional form is isotropic. And you just sort of have to get your hands a little bit dirty by thinking about congruence class is not eight and so forth. Okay, sorry. So there was a question. Yes, that's right. And so, right. No, so, right, sorry. So the key feature of any local field is that there's a unique an isotropic four dimensional form any any non Archimedean local field. Right. So the you invariant is foreign. There's a unique example of an an isotropic four dimensional form. And in the case of Q two, you can write it down explicitly. It's it's this particular form. Yeah, so I'm going to put this on the problem set for today. Okay. So sorry. So there's a question. Is it difficult? Right. I mean, I guess so hence was almost so true. It's just that when you when you when you have squares, then two is not a two attic unit. So it's not true that anything congruent to one mod two is a square you need to be congruent to one mod eight. Okay. But so just to go back. So there is a description. There is a description of the bit group of any of any non Archimedean local fields. So up to this filtration. So although you don't have this nice description as it's a direct sum of the bit group of FP plus a direct bit group of FP, when P is greater than two, you always have this finite filtration. And you always have that I cubed is equal to zero. And you always have that the associated graded terms are Z mod two, E cross mod E cross squared and Z mod two. So sorry. So just to recap, the bit group of E is a local field has a filtration where the associated graded terms are w of E mod I is given by Z mod two. I mod I squared is given by E cross mod E cross squared. So for example, you work, I mean, I think in the exercises, maybe in Friday's exercises, there was exercise of working what this is, especially when he is Q two. And then I squared mod I cubed is equal to Z mod two. So you don't have a just you don't have a direct description as, as an abelian group of the bit group of E as like a direct sum of two things. But you always have this, you always have this sort of three step filtration. Yeah, sorry, question. So I was wondering, looking at the filtration, if we think of it, if we think of all those things, not as well, rings, but just as abelian groups, and we think of this as a, as a actually as a normal, as a normal series. Well, in general, this thing is not going to be a composition series, right? Because this E star modulo E star squares and going to be simple. Is there any, like in the case of QP, for example, for P greater than two, we have Z mod four Z. Is there any interest in well, extending this to to a composition series? Like, is there any interest in treating this as a as a normal series and extending it to a composition series? Sorry, what do you mean by the difference between a composition series and a normal series? Well, I mean, in this case, because all the groups are abelian, I guess that well, subnormal and normal, they coincide in this case, right? So I'm just I'm just looking at this filtration and thinking of it as basically a sequence of abelian groups and not just rings. And I'm wondering, well, this, this factor, this E star modulo E star squared, that doesn't have to be simple. But of course, we can refine. If we look at it as a series of abelian groups, we can refine it to a composition series. Would there be any, any information hiding there? Like, is that something that's investigated in, in the bibliography? So at least if I understand your question, I think it's not going to be very canonical, I think it's not going to be very canonical because these are F2 vector spaces. So there are many different ways. So I mean, E cross mod E cross squared is going to get quite large when E is like some big extension of Q2. But so I think what what really is what one really wants to do is to understand the multiplication. So the thing is that like the exact, right, so I think is what I think what you said might not be the composition series is not necessarily going to be very canonical. But what you do have so that there is some extra structure here, which you do want to understand. So, so this is right. So this is this is as an abelian, what I've described is, is at the level of abelian groups. And so, so I cubed is zero. So, so this is a sort of a complete sort of filtration as abelian groups. But there is a little bit more information here that you want to understand, which is a multiplication. So the multiplication, right, so there's a map from i mod i squared times i mod i squared into i squared mod i cubed, which is also just i squared, which is the mod two. And so, right, so if you want to understand the the bit group, you want to understand some of the multiplication. It's, it's given by, well, you can try to understand it by looking at the multiplication on associated graded terms. And so this is a pairing, which goes from E cross mod E cross squared times E cross mod E cross squared into Z mod two into plus or minus one this this multiplication law.  Sorry, so question. So that's a Z mod two. Yeah, it's not, not, yeah. Okay, above, sorry, above where? Yes, like your multiplication and E square and then what's the other term? So it's i mod i squared. Sorry, you mean here? Yes. Yes. Sorry, it's the same thing. It's i mod i squared. Thanks. All right. Okay, so, so, so there's a natural pairing that this gives you and this pairing is actually sort of really fundamental for studying quadratic forms over local fields. And it's, it's this construction which is called the Hilbert symbol and it's sort of a general feature of local fields. So definition, let E be a local field. So given elements a comma b and E cross, we're going to define the Hilbert symbol brackets a comma b to be the following. So it's plus one if z squared equals a x squared plus b y squared has a non-trivial solution. So ie if brackets one minus a minus b is isotropic and it's going to be minus one otherwise. So this is called the Hilbert symbol. And it's just a sign. So the Hilbert symbol is just a sign that you can associate to any elements of, of E cross. All right. So, so for example, so, so maybe let me list some, so just have a few minutes off. So maybe let me just list some, some, some properties of the Hilbert symbol. So explicitly where this is coming from is that the Hilbert symbol of a comma b, so sorry since I've been talking about bit rings, is the following. Well, you form, you consider brackets one minus a, tensor brackets one minus b. And so this, this lives in the bit ring of E. So this, this lives in I, and this lives in I. And when you multiply them, this lives in I squared, and I squared is exactly z mod two. And so the Hilbert symbol is exactly, is exactly the sign that you get out of this. So brackets a comma b is equal to, well, you can, you can equivalently say it this way. So it's, it's one if the form brackets one minus a minus b, a b over e is hyperbolic, which, well, you can check this is, so I'll put this on the, on the homework that's the same as brackets one minus a minus b being isotropic. It's the same as this being hyperbolic. And it's minus one if the form brackets one minus a minus b comma a b is an isotropic. And so recall that there's exactly one choice of an isotropic form of dimension four over a local field. Okay. So, so, so, so if you have elements a and b in your in, in the group of units of your field E, you cook up either this three dimensional quadratic form or this four dimensional quadratic form. And then you ask whether it's either in the first case, whether it's isotropic or not, or in the second case, whether it's hyperbolic or not, sort of equivalent. And then that, that is encoding this construction called a Hilbert symbol. So the Hilbert symbol is right. Okay, so yes, so there's a question, why, why does E need to be a local field? Why, why can't we do this just very generally? Right, so we can do this very generally. But it's not necessary. So you can make this at least the first definition or we can make either these definitions very generally. But it's not going to be a very good definition for the following reason. So, right, so let's, let's, let's write, let's write down some, some, some properties of the Hilbert symbol. So the Hilbert symbol is going to satisfy various properties. So for example, the Hilbert symbol of a comma one is always one. Because, well, right, the Hilbert symbol of a comma one minus a is equal to one because you can always solve z squared equals a times one plus, well, because you can write one is equal to a times one plus one minus a times one. A comma minus a is equal to one. So, so you can write down a bunch of basic properties of the Hilbert symbol that do not use the fact that E is local and you can, you can make this definition. But what's actually kind of sort of special about the case that E is local is that this is bilinear. And so it gives a bilinear pairing, it gives a symmetric bilinear pairing from E cross mod E cross squared times E cross mod E cross squared into z mod two, into plus or minus one. And in fact, it's non-degenerates. Okay, so maybe this is where I should sort of, I should sort of wrap up. So, so, so what's key about a local field? So I'm going to have to come back to this next time. What's really key about a local field and about this construction of the Hil, of the Hilbert symbol is that by looking at whether or not you have a solution to this quadratic form, you get a bilinear pairing from E cross mod E cross squared with itself into z mod two. And this bilinear pairing is actually non-degenerate. So I think in, you know, I'm going to try to put this on the problem set, but this, so you can, you can try to work this out very explicitly for q p. And when p is greater than two, this is going to give you essentially the quadratic symbol or the Legendre symbol. And for q two, it's going to be a little bit more complicated. For q two, of course, it's going to, you're going to have to try to work out some conferences mod eight. But right, so, so I guess the main thing that I, you know, sort of wanted to explain is that you can think about this in terms of the vet ring. So this Hilbert symbol is something that is encoded in the vet ring, because it's exactly the map from i mod i squared times, it's exactly the multiplication squared, the multiplication map from i mod i squared times i mod i squared into i squared mod i cubed, which is z mod two. And i mod i squared is E cross mod E cross squared, i mod i squared is E cross mod E cross squared. And this, this sort of multiplication law in the vet ring is exactly the Hilbert symbol. Okay, so sorry to end in, I say the awkward place, but yeah, so thanks. I'm going to, I'm going to pick this up tomorrow and say more about that. So stick around for the questions. Is there, okay, can I ask a question? Yes, please. So is there a reason why, like, even dimensional, the ideal of even dimensional forms is important and like not like dimension multiple of three? Oh, sorry. Oh, that's because, okay. Yeah, exactly. Right. Yeah, sorry. That's, that was obvious. Yeah, sorry. So the question, there's a question. Do we need E to be a local non-archimedean field of characteristic not two to define the Hilbert symbol? So the Hilbert symbol, sorry, so I guess we need the characteristic not to be not two. But in fact, yeah, right, so you can define the Hilbert symbol over the real numbers as well. So, so this, this, this definition that I wrote up here of the Hilbert symbol, it works over any, any local field of characteristic not two. And it's going to be, it's going to have this bilinearity property. So, right. So for example, you can define the Hilbert symbol over the real numbers. And then it's going to be minus one, if and only if A and B are both negative. If you're over the complex numbers, it's just going to be, I mean, C cross mod C cross squared is zero. So there's nothing there. But yeah, right. So yeah, so that's, yeah, that's a great question. I mean, that's exactly the, sorry. So the question is, you know, what does this have to do with Galois cohomology? I mean, this is exactly the subject of the Milner-Conjecture, the theorem. So, so in fact, the, the, so I will try to tie this together next time. But since I'm not going to say much about Galois cohomology, in fact, what happens is that for any field, so we can write this for any field. So, right. So for any field F, one considers this iatic filtration, one considers the bit ring of F, and one considers this iatic filtration. And so in fact, the, I guess the Milner-Conjecture, so a theorem of, in this case, Orlov, Rijek and Riewadzki, is that if you take this associated graded ring, so after that ring, so this is the direct sum of i mod i to the n mod i to the n plus one, then this is isomorphic on the one hand to Milner k theory, mod two, which is what I would talk about tomorrow. And also to the Galois cohomology of the field F with coefficients in, yeah, C mod two. So in fact, that's exactly, exactly that. Well, I guess that's, so there are actually two Milner-Conjectures, and I think that the relation with Galois cohomology was the, was, yeah, was proved by Riewadzki. And maybe I think it was proved first. Right, sorry. So the question is, right, so how do we extract information about a ring using knowledge about its filtration? Right, I mean, I guess it's an example you would give in an algebra class. Um, I don't know, I mean, right, so if you have a, I mean, if you have a filtration on the ring, it doesn't quite determine, it doesn't, if you know the associated graded ring, it doesn't quite determine the ring. But, I mean, for example, it will tell you, I mean, it will tell you how to compute a product modulo terms of higher filtration. So I don't know if I have a great answer, I don't know if I have a great answer to that question. Um, I mean, sometimes, right, I mean, so for example, if you know, if you want to show that a product is nonzero, then if the product is nonzero and the associated graded, then it's nonzero in the ring to start with. But for example, it's going to be hard, if you just know the associated grade, it's going to be hard to tell whether something actually is zero to prove that something is zero. Because the associated graded is going to let you prove that something is zero modulo higher filtration. Thank you. Thank you. Okay, so I'll see you, see you tomorrow or at the office hours.


Hey, what's going on guys and welcome back to another video. So I'm back with another Shopify store review from Flipper.com. Today we are looking at Lux Strap. It's currently for sale for 60,000 US dollars. And in the grand scheme of things, it's quite a modest Shopify dropship in store. When I say modest, I mean they're not making millions of dollars. They're not even making hundreds of thousands of dollars. They are making on average about 24 grand profit per year. I wanted to show this store on my channel because not many people talk about businesses of this size because they're not really, really exciting. However, from the majority of people I actually talk about when we're striking up conversations and just getting to know why people want to start dropshipping in the first place. It's not because they want to become millionaires in the next six months. If that is you, then this probably isn't the video for you. However, if you're looking at getting started in dropshipping because you want to make a few hundred extra dollars each month or even an extra couple of grand each month, like these guys have, then this is exactly what a business of that size looks for. And I'm very confident in saying that everybody watching this video is capable of achieving and replicating a similar level of success to these guys because in the bare bones of it, this is a very basic business. Now, that's not me taking shots of this business far from it. The majority of people who try dropshipping, the harsh reality is most people don't make a single penny of this. So for these guys to make a consistent, the profitable business over the course of the last two years is by no means an easy thing to do. So for some more information, then it's called Luckstrap. We'll be taking a look at their Shopify store. We'll be taking a look at their best selling products. I'll show you where you can sell those products from as well. And I'll also show you how they are advertising those products, too. So if this is a niche that particularly captures your attention, you want to create something similar, I'm going to give you all of the components to do so. So Luckstrap, I've already mentioned about half a dozen different times now. This is a verified listing. So the numbers are taken direct from the source of supply. They are not editable. Everything you're going to see in this video is legit. So the size age is two years. The monthly profit here is 1400 US dollars. However, this is actually off. I will show you the numbers in a second from the last 12 months. This is still an active business. These guys are still actively making over two grand a month in profit every single month. So this is actually low. So the asking price for this business is $67,000. I think that's a bit high, to be honest. And when you value in a store or if you've got a relatively successful store and you're looking for an exit strategy and you want to sell your store, you're looking at about 12 months to 18 months worth of profit. So if we apply that to this business, then you're probably looking at about closer to 40 or 50 grand. So the key highlights of this business, I won't bore you for too long for this information. However, I believe it's important, especially if you are a beginner looking to get into dropshipping, you should find it interesting or you should at least want to know what goes into running a profitable online business. So a profitable online business in a rapidly growing niche, specializing in premium, Apple and Samsung watch bands, cases and accessories. Primary revenue comes from sales of watch bands, cases and accessories, which I've already stated. I'll show in a second how it's primarily just watch bands. The business requires an estimated time commitment of 10 to 15 hours per week to manage inquiries, order processing, inventory management and marketing efforts. So whilst, say 25 grand a year is not a lot of money for some people, when you actually pro-router that into the time invested into the business, it's a pretty decent return on your investment. Most people, I would say, as long as you haven't got a large family can probably find 10 to 15 hours per week to build a business of this size on the side. So if I just show you the primary expenses, the breakdown on the monthly basis just to illustrate how the numbers are off, however, what numbers aren't off are these here down below, which we'll take a look at in a second because these are pulled directly from Shopify. So shipping 1600 a month, that seems like a lot of money to me. This to me is coming across as a business who has a really solid product and customer base. However, they seem to be spending a lot of money on the things that they're doing. So maybe instead of trying to focus on increasing their revenues, maybe decreasing their costs could be a great way of increasing the profitability of this business. So marketing about five grand a month cost of goods sold 1500. So how they're spending the same on goods as they are on shipping. I'm not quite sure unless that's an import mistake. Platform for fees 336 discounts 600 SEO 1400. I'm not quite sure how they're spending $1,400 on SEO each month and whether they're seeing a return from that or not. It'd be interesting actually to have a bit more information on that. Okay. So they're revenue and profit them from the last 12 months. So of April last month, I record this in May. So it's still current still an active business. They're still actively running ads, which I'll show you in a second. They turned over $13,000 and made just over $2,000 in profit. I'm not quite sure how their revenue is so high with their profit being so low. But again, without that information, I wouldn't be able to tell you why the key takeaways from this is we can see that it is a product that has potential and it is a product which they've consistently been able to turn over and make a decent amount of profit each month given the time and effort they're putting into it. So I've seen the numbers then. Let's jump on to their Shopify store and take a look at exactly what their store looks like. So look strap. As you can see, they're going for a more kind of expensive look and feel with the kind of calligraphy type style fonts, which simply says their names. So a nice touch of big background. This looks like the old school Brooklyn theme. I'm just going to check out what theme it actually is. Some sort of custom theme starts with thunder, upgrade your wrist, different collections. So as you can see, there's a heavy, heavy push here on the watches and not so much on phone cases or any other type of accessory. But all in all, like a pretty default template style store, I don't think they've done too much work to this theme other than actually import and upload their products. What is slightly conflicting for me is they go for that kind of luxury branded type message that they're putting across. But if we actually have a look at their products, they're $30. That's not a luxury product in my opinion. So there's a bit of kind of cross, crossed paths or a cross message being put across there. And I think some kind of like the brand could become more uniform and in tune with the type of products they're selling, then maybe they'd be able to make it that bit more profitable and bit more of a success. So we've seen the numbers. We've seen what their Shopify store looks like. Let's take a look at what their best selling products are. So unfortunately, they are hiding their best selling products. However, what we can do is we can jump on to the Facebook ads library and we can see exactly what products they're advertising. And what we can see is they've launched two different products in May of 2023. We have this scrunchie style Apple Watch Band. I don't know why I forgot what that was called. Obviously aimed more towards a female market, I would guess. But nonetheless, kind of like quite a unique product. They have it in lots of different colors and patterns and styles and this looks like a dynamic ad to me, which basically means Facebook will show that to some people, will show that one, will show that one, will show that one to some people, see which one gets the best results and then just continue showing the better performing one. The other style that advertising is a more male targeted product, in my opinion, so we can see the details of this one. And again, they have lots of different variations of this particular product with the leather strap heading the ad. Let's jump on to the product page then and take a look at what their product page setup is for this particular product. As we can see, like the logo is displayed a bit differently with an outline to make it a bit bigger and bolder. It is indeed the same website. It's got the same URL, but it kind of looks different to me. Again, it's a cheap product. It's $25. It's definitely not a luxury product and they're even offering people to break that down further and pay for it for as little as $10 per week. Something else they're doing, which is becoming more increasingly popular with a lot of successful Shopify stores, is this bullet point feature section, I guess, or bullet point benefits. I'm going to call it. Something I've started doing for some of my products as well. And something more increasingly important is making it as quick and easy as possible for a consumer to make a purchase to give them all of the information without them having to go hunting for it. And this is where these bullet points really do come into play because when that product page loads, if they have all of the information there and then with a big A to C button, that's going to convert more than having some of this information at the very bottom of the product page where they may have to go looking or searching for or even go on to a different page. So by having all this sort of thing at the top here, it does a really good job of really summarizing the product. Given the consumer, all of the information they need to make a trustworthy purchase. Something I also recommend all Shopify stores to have is this delivery estimate. Again, think about it. When was the last time you bought anything without knowing when it's going to arrive?  And what's actually really interesting about this product page is if we scroll down, reiterating those same bullet point points, they've got the recommended, they've got a bit of like a section about the product at the end. And then they actually show a completely different product. And then they actually have kind of like two product pages on one, which is very strange to see. It's something I've never seen before in a Shopify store, but it's obviously working for these guys. So who am I to criticize? Maybe that's something that you would split test and change in and out week and week just to see how it affects your conversion rate. So I think in the grand scheme of things, it's probably got a validated idea here that works. I think it just needs a bit of optimization, let's say to really kind of hone in the best strategy, best product page, best ads and best source of traffic to really bump up those profit margins. The next product page is for the lever product. So again, it's not a premium or expensive product at 30 pounds, especially if it is legitimate leather. The product page by no surprise carries on that same strategy and theme. They have the buttons for the different variants, rather than drop downs, which is super important in my opinion. They have the same delivery estimate, some trust badges, different tabs, that sort of thing, a size chart. So the consumer knows exactly what size they need. And again, with the strange product layout, with the different products below and an actual completely separate different product page, it's very strange to me, however, that I'm sure they have their logic and their own reasoning for doing this. So we've seen the level of success in terms of the numbers and the contextual information, we've seen what their best selling products are and what their ads are. Just to kind of point out these are just very generic single image ads, very easy to replicate if you did want to go ahead and sell the same products. I'm sure these will just be images taken from the supplier. We've seen exactly what the product pages look like. Next, I'm going to show you where to source these products so we can get a feel for the types of profit margins these guys are having. So $25 for this particular product. I mean, if you're going to Aliexpress and search scrunchy Apple Watch Band, as we can see, there's endless suppliers selling endless different color schemes and variants for this particular product. But what is fairly consistent is the price. So they're all under two pounds. That does include free shipping in some cases. So here's one particular supplier found so free shipping 15 day delivery on time guarantee for 40 pence. And as you can see, lots of different color variations to give your consumer that option. I even found a like for like. In fact, I think that's the actual image they're using for that ad. I'm pretty confident that it is. So that is one pound 66 plus one pound 59 delivery. What's that about less than four pounds delivery? So definitely 20 quid or 20 pounds worth of room in there. So we're able to set aside enough room in the retail price versus what their cost of goods are to dedicate towards their marketing. As for the leather strap. So if we go back and just refresh our memory, so 31 pounds is a slightly different one. I'm not going to bore you with showing all the different variations and suppliers for this particular product. Here's kind of like a universal supplier that I found that sells all types of different sizes. But if I'm not mistaken, if we compare the images, then it's the same type of lever with the same type of design. So for all intensive purposes, an identical product. So for this, you can source it for three pounds just under with a 15 day delivery on time guarantee. They're selling it for over 30 pounds. So again, a nice cheap product with a decent profit margin, which leaves a bit of room in there for scaling as well. And so with that being said then guys, that is the full breakdown of looks strap com. Like I said in the beginning, a modest Shopify store, but they clearly found an idea that works here and whatever they're doing is working for them. I think it just needs a bit more care and attention or time invested into the business to test things, to split test things, do a B test in to really kind of focus on those most profitable traffic sources, those better maybe optimized product page layouts to really help increase and bump up the profit of this store. But hopefully if you're still stuck with me, you found it useful. There's some things I hope you can take away from this video, which will help you build your business and be able to replicate a business similar of this size and go on to exceed this. Thanks for watching guys. Any comments, questions, video suggestions, anything at all. Post them down below. I want to try and help people out as much as possible. So make sure you span the comments section. I read every single one. So we'll get back to you. Thanks again. I'll see you in the next video on Wednesday. Cheers guys.


فَبُتْطُّهُ آدَكَ الْأَيَّا مُفَتَّى وَتَنْهِ لذا نقول جاء موسى موسى مرفوعية جاء موسى موسى هي مرفوع رأيت موسى موسى هي منصوب مررت بموسى موسى هي مجرور ولكن نقول تقديرا العلامة of it is مقدر يعني our eyes can't see it it is not apparent so those are the three points that I still have knowledge has to observe هو التغيير أواخر الكلامي لاختلاف العوامل الداخلة عليها here the last one is وأقول I will say محمد محيدين عد الحميد وأقول I will say الإعراب الله معنى يعني إعراب has two meanings أحدهما لغوي one is a linguistic definition والآخر صلاحي and the second one is a technical definition the technical definition according to the grammarians أما معنى هو في اللغة as for its meaning in the linguistic meaning as for its linguistic meaning is أل إظهار it is something to be apparent والإبانة the word إعراب linguistically means أل إظهار والإبانة إظهار and إبانة both mean the same something to become clear something to become apparent the إعراب means for example the Arabs they say أعربت عما في نفسي أعربت عما في نفسي means what I have made apparent that which was within me are you with me أعربت عما في نفسي I have made it apparent that which I had inside my heart everyone now knows it أعربت we are looking here is أعربت and that's it there is another meaning it has that the sheikh didn't mention but we are going to come to it إن شاء الله يتعالى إنا ومتميمة الأجرومية إتغير changing أعرب means also changing for example you say أعرب الله الله changed معدة my stomach or the way my stomach is it got changed it hurts now the Arabs use that as تغير as well على كل حال الإعراب linguistically it means الإظهار والإبانة تقول أعربت عما في نفسي إذا أبنته إذا أبنته وأظهرته إذا أبينته what did you say if you make it clear if you make it apparent and you make it clear it's called what it means أعرب أعربت معما في نفسي I made it clear that which was with me وأما معاه في الاستلاحي as for its لسه ذاست اللي linguistic meaning as for its technical meaning it means what هو ما ذكره المؤلف is that which the author has mentioned that's the technical meaning which is what هو التغير أواخر الكلمي لإختلاف العوامل الداقيلة عليها لفظا أو تقديرا now we're going to stand over each point we're going to go through each point that the author said even though I did it for you overall but now we're going to stand over it properly والمقصود من تغيير الأواخر الكلمي the intent of the meaning of the changing of the last word is تغيير أحوالي أواخر الكلمي it means the changing of the situation of the last wording وَلا يُعُقَالُ because somebody might think the changing here means what he might think that the دال is going to change it to a راء are you with me somebody might think to himself it might be the راء the دال might change it to a راء so we have to keep saying it is تغيير أحوال the situation أواخر الكلمي وَلا يُعُقَالُ and no one should or it shouldn't be understood as and you rather تغيير نفس الأواخري that the literal ending changes فإن آخر الكلمة نفسه لا يتغير because the last ending of the word itself doesn't change وتغيير أحوال أواخر الكلمة عبارة عن تحولها من الرفع إلى نصبي أو الجرري the changing of the situation of the last ending of the word it means the changing of it from بن مرفوع to منصوب and to مجرور حقيقة whether it's literal أو حكمة or whether it's by way of ruling ويكون هذا تحول بسبب تغيير العامل and that changing of the ending of the word is the reason is because of the situation changing من عامل يقتض الرفع على الفعالية أو نحوها like the factor of making it مرفوع because of it being a فاعل for example if I say to you for example now جاء زي دن who caused the zaid to become مرفوع you say جاء because if جاء never came with زي دن ever be مرفوع it would never be مرفوع as a فاعلية here are you with me what caused it to be what what caused it here the عامل that caused it here is جاء at the فعل is جاء at the فعل so what causes a فعل sorry something to become either a فاعل or a مفعول بي or whatever it is it's a عامل it's a factor are you with me من عامل يقتض الرفع على الفعالية this is the فعل أو نحوها or like it إلى آخر يقتض النصب على المفعولية who forced it here to become what for it to become a مفعول بي is that if the verb if the verb it becomes what it requires a مفعول بي for example جاء زي دن does the word جاء require a مفعول بي no because it's a فعل لازم it's not a فعل متعدي it doesn't require a مفعول بي like in ضربة what does it require ضربة زي دن can I just stop there no because ضربة is a فعل that requires a سفع المتعدية so wherever the verb is it necessitates the مفعول بي either to come or not to come so that's عامل نعم it's عامل أو نحوها وهلما جراء and let the list go on it can still go on okay it goes on now the sheikh is going to give an example so before I move on from that فقرة from that paragraph what the author is saying is what I said before the changing here brothers it's that's not the changing literally the دال or the لتح the حرفة the حرفة that's there it means the situation is what's causing it to change okay that changing is either by way of ruling or literally it changes and the reason why it's changing is because there is عامل a factor that's making it change that factor may force it to be a what a رف or it may force it to be a نص or the list can go on okay مثل an example إذا قلت if you say حضرة محمد المحمد came فمحمد المرفوع محمد is مرفوع here why is he مرفوع لأنه معمول لعامل يقتض الرفع على الفاعلية وهذا العامل وهو حضراء because he is a factor is making it to become a فاعل which is the factor is who who's forcing it to become it's حضرة حضرة is forcing it to be are you with me good فإن قلت if you say رأيت محمد I saw محمد تغير حال حال آخر that change that ending will now change if you say رأيت محمد محمد is not it's not محمد now it's محمد why it's changed to what إلا نصبي it's changed to becoming نصوب صحيح it has it has why was forcing it to change لتغيير العامل بعامل آخر يقتض النصب وهو رأيته the reason why it's now changed is because the عامل has changed the factor has changed which necessitates it to become نصوب the عامل here is رأيته رأى which is فعل ماضي the تا which is a ضمير متصف في محل رفع فاعل the tab by itself is standing in the position of a subject so if you see رأى does it require an object of course it does it's not a فعل لازم it's a فعل متعدي فإذا قلت if you say حضيته بمحمدن حضيته بمحمدن it changed from what it changed from مرفوع and it also changed منصوب to what مجرور because there is عامل that forced it to be مجرور وفوست محمد here right now to be مجرور تغيير حاله آخره إلى الجر why لتغيير العامل بعمل آخر يقتضل جر وهو الباء الباء here is what's forcing it الباء is عامل that forces the noun to become what مجرور you good وإذا تأملت if you observe a student of knowledge في هذه الأمثلة in these examples ظهر لك it becomes apparent for you أن آخر الكلمة the ending of the word وهو دال the dal من محمدن keep looking at the dal which is the last word لم يتغيير it hasn't changed so the dal does not change it's never going to be باء or جي محق أخاء it's always going to be دال but what changed وأن الذي تغيير but what changed هو أحوال آخره the situation of his changed فإنك تراه مرفوع because in the first example which was what حذر محمدن you find that it's what مرفوع that is مرفوع في المثال الأول ومنصوبا في المثان الثاني in the second example it is what منصوب رأيت محمدن ومجرورا في المثال الثالثي and the third one is حظيت محمدن it's also it changed because it became مجرور وهذا التغيير وهذا التغيير and that changing من حالة الرفع إلى حالة النصب إلى حالة الجري from the situation of رفع to the situation of نصب to the situation of جر هو الإعراب that's called إعراب عند المؤلف according to the أوثا ومن ذهب مدهبه and those who have taken his مدهب there's another مدهب what is the other مدهب what are the two مدهب that are out there some people believe that they believe that إعراب is something which is what معنوي it's not literal and some believe it is literal some believe that it's literal and some believe it is it is not literal those are the two مدهب that stand those other those other two مدهب that stand وهذه الحركات الثالث التي هي الرفع والنصب والأمارة على الإعرابي so what do those people say are you with me who believe it is not it's معنوي those people who believe it's معنوي who don't believe it's literal the author here he believes it is literal he's saying to you the author is saying the رفع itself is the إعراب the نصب itself is the إعراب and the جر itself is the إعراب what about the other ones who say it is not they're saying that the رفع والنصب and the جر are only signs for the إعراب but that's not the إعراب are you with me so if you look at their dispute what is it called اختلاف واللافضي اختلاف والتنور their differences doesn't have a real impact it doesn't really have something there's a خلاف where when you look at it it makes the two groups go different directions and they will come out with different results that's not what happens here they both still on the same road it's just they see it means different so this خلاف is called خلاف واللافضي it doesn't really have much weight but but إن شاء الله متنممة الأجومي we will go into it a bit more we will go into it a bit more والنصب والجر هي علامات وامارات على الإعراب ومثل الإسم في ذلك الفعل المضارع فلو قلت يسافر إبراهيم فيسافر فعل ومثل ومثل ومثل الإسم and the example of the إسم is في ذلك الفعل المضارع because all this that we took right now is who now we kept taking the word محمد which is an إسم the likes and the example of the noun is like the فعل المضارع فلو قلت if you say meaning the فعل المضارع it changes from situation to situation let's take the word for example يسافر إبراهيم تقوموا بعمل المضارع. ماذا تفعل؟ ماذا تفعل؟ ماذا فعل؟ فعل مضارع. كيف تعلم أنه فعل مضارع؟ نعلم أنه فعل مضارع بسبب ماذا؟ بسبب أنه إذا نحاول أن نضع مضارع فيها، سنستطيع. هل سوف تفعل؟ هل سوف تفعل؟ فعل مضارع. هل أنت معي؟ because the scene and the sofa were uniquely only those who entered on to what? فعل مضارع. فأن احاول أن نضع مضارع، فهذا سيستطيع. فهذا فعل مضارع. فهذا يسافروا. فعلا فعل مضارع. مرفوع. إنه مرفوع. يسافروا. فعل مضارع. ماذا تدفع؟ أنه مضارع الان بسبب أنه يستطيع أن نضع مضارع before it. هل ستفعله؟ نعم لأنه سيفعله ، نعلم that the only one they accept seen and so far is who فعل مضارع the present verb good so it's مرفوع it's مرفوع because look at the ending of the letter or the word دراء دراء is مرفوع يسافره that's مرفوع why is it مرفوع؟ لتجروده من عامني يقتضي نصبه أو عامني يقتضي جزمه the فعل مضارع only goes through three situations and I don't want to complicate it too much on to you but it goes only through the now the verb the فعل مضارع goes through three situations one it is what? it is either رفع أو نصب أو جزم the رفع is when there is nothing before it when there is nothing before it what do I mean but there is nothing before it meaning there is no word there is no حرف or there is no adat that go before it that change it from either نصب to جزم to نصب or جزم are you with me? so the فعل مضارع goes through three situations الرفع والنصب والجزم did I say جر؟ جزم very good there are two times that it becomes نصب and جزم it's easy because you just need to learn the words that go before it that make it منصوب and they call it أدواتو نصب you just memorize them and the next time it becomes مجزوم you just need to memorize أدواتو جزم that go before it and it becomes مجزوم whereas the فعل مضارع when those two are not there the grammarians they say لتجرد تجرد means what? when you tear something off something or you strip it off it so the فعل مضارع here it's stripped off from what? أدواتو الجزم and أدواتو نصب it's free from it so you suffer is there anything before it? no sorry you suffer is there anything before it? from أدواتو الجزم or أدواتو نصب no there isn't so what we say here is مرفوع فعل مضارع مرفوع لتجردوده من عامل يقتضي نصبه or عامل يقتضي جزمه فإذا قلتا for example if you say لن يسافر pay attention now the word يسافر has something on before it now the word لن went before it right then لن is from the عامل يقتضي نصبه it's عامل it's a factor it's a thing that forces it to become what نصب for example look at the word يسافر لن يسافر you say here when we look at the word يسافر we grammatically analyze it we will say يسافر is منصوب وعلامة نصبه الفتحة it's منصوب why? because the عامل of أدواتو نصب one of them entered it the word لن whenever it goes before فعل مضارع what does it do to it? it makes a منصوب so here now what happened تغيّر العامل the عامل changed here from now what did it change from? it changed it from it's what it used to be which was رفع and it made it into مصوب good instead of لن put the word لن before it الفتحة مضارع so don't say لن so this author said look و هو لن فإذا قلت لن يسافر keep looking at the راء because that's what you have to do إعرابوزوات والتغيير الأواخر أواخر الكلمة look at the ending of the word the راء what happened to it here right now? what happened to it here right now it became مجزوم why did it become مجزوم? because the word لن went before it and the word لن when it goes before اي فعل مضارع what does it do to it? it makes it مجزوم so this is an عامل a factor a thing that forces it to become what for it to change so تغيّر حالو the word يسافر it used to be and then it changed to يسافراء and then now it changed into what you say فر the reason is because the reason why it changed is من الرفع أو النصب إلى الجزم لتغيّر العامل بعامل الآخر يقتضي جزمه وهو لم the word لم forced it to be


Yes. Okay. Um, remember when I remember I was shouting out I was pronouncing her name wrong cause I'm boning but it's bonon bonon Remember I was saluting her and saying how I follow her on Instagram and she's just Living like life.  This is bonon Okay, nice to meet you. Nice to meet you.  Okay, so Which is random come in New York, but there's all these like iconic surf spots. I want to go see in South Africa Jeffrey's Bay is the big one. That's the bit. That's the big one. Yeah, Plattenburg Bay Plattenburg Bay is But the thing with Jeffrey's Bay is like there's all these great white sharks yes, is that something that you guys are concerned about when you're swimming that like Of shark attacks. Uh, yeah, especially particularly in Cape Town around Musenburg.  I mean, no No, why do you think we do that? Why do you think we do all those crazy things because you have money to do those things? Yes, but you have money.  You're not hiking. Don't blame that on money I think I think it has something to do with like when life gets boring we need to feel like we have to survive Okay, wonderful, you know like I don't think that like you know There's parts of the world where people are really struggling and I don't think they have the same desire to skydive Yeah, right because every day is the struggle every day is survival. It's like find water find food Yeah, that's survival. But when life is too easy, you start getting bored. You're like, okay, I need to go snowmobiling You know, I need to put myself in danger I need like so many people go to go to Africa and they go on safari right and like they actually they'll go hunt animals Right and it's like I'm like, what is that about? Why do you feel the need to like tap into your like Primal roots and I was talking to buddy of mine He was like I really think that like these are like reward systems baked into our humanity It's crazy, you know like going out there like why are fewer worth like millions and millions of dollars? Are you taking a gun in the middle of the forest and trying to shoot?  It's got to be expensive to do that a private. I'm sure it is I wouldn't really know I've seen people we see people coming.  I'm sure is it is it annoying to answer like animal questions When you come to the States all the time. Yes Well, don't walk around in chinchillas.  All right Don't walk around animals on you Yo in South Africa it's since 1905 it's only been 248 Unprovoked shark attacks only unprovoked.  Yeah, so you have to come and surf in South Africa. You won't get eaten by a shark See, no, I would love to love it. The water is warm. The people are warm. The food is great Yeah, the weather's great. Yeah, everything's just fine. Okay now now another question I have languages. Yes. Okay There's obviously everybody speaking English. There's 11 in South Africa.  Look at this manscape razor right here. This is for one thing It's not to trim your mustache.  All right That's what we want to do.   Well, I'm trying this and the reason I'm trying this and I've been telling y'all for Long as I really don't want no hair on my boss. I don't want to have my pubes. I really don't we got to get rid of it I didn't want any be women with the ball vaginas. I don't want nothing. I think we got to start doing this This is non-stop. Maybe we even look into a laser situation Listen, you can't hurt yourself because they got this clip on It's guaranteed. Anyway manscape.com use our promo code. Make sure you go get it. Peace Okay, so you speak interest English Afrikaans Afrikaans is like a blend with like Dutch or German. Yeah, German Dutch I think I believe so it's got a lot of us and yes Yes, yeah, it's like Arabic And what else is so Anna so too northern and and southern so too are two different languages The clicks That's easy Yeah, so there's 11 official languages in South Africa, so but everybody unites usually around English English and Yes, okay.  Why do you think Americans have this? You have a New York accent I got kind of the down south list I guess a little So just everyone speaks English yeah like you have different neighborhoods that are gonna have different in different actions different areas But but I do think that there is this kind of like universal like literally isn't is everywhere, right? Yeah, a girl in New York could say that so there's something cultural cultural about America that like we have to Life is so good. We have to like prove it's real Oh This is great observation why are we constantly trying to prove these things are really America's I think on the low America. No, we kind of foolish it we do we kind of know that we foolish it just a little bit Because the things that you're gonna see right in South Africa like they require literally way more like literally a great white shark ate That white kid literally that needs a little Literally it's 30 degrees I must tell you after all your seal all this black man's I'm looking at you Yo, yo, yo your patch cuz I'm thinking it's a brother up under there who and I have a brother up under me What would that make him if you had a brother up under him