Next to capacity and effect of consent, the last reason for invalidating a treaty is that its very object is illegal because one of its obligations conflicts with a fundamental rule of international law. Under Article 53 of the Vienna Convention that I quote, a treaty is void if at the time of its conclusion, it conflicts with a parametry norm of general international law. For the purposes of the present Convention, a parametry norm of general international law is a norm accepted and recognized by the international community of states as a whole. As a norm from which no derogation is permitted and which can be modified only by a subsequent norm of general international law having the same character, end of quote. In the title of Article 53, parametry norms of general international law are called in Latin Yuskogens norms. Let us first try to understand the very notion of Yuskogens before turning to its normative status, the ways it is formed and finally give some concrete examples of parametry norms. As such Yuskogens is opposed to Yus dispositivum. What does that mean? Well simply that some norms can be derogated from by agreement between contracting parties and those norms are called a Yus dispositivum they can be freely disposed of. While other norms cannot be set aside by mutual agreement and those norms are parametry they are Yuskogens. This dichotomy exists in every domestic legal system. Some legal provisions apply in the absence of any particular contract while other legal provisions apply despite any contract and contracts that do not conform with those parametry provisions cannot be upheld in a court of law they are invalid. Parametry norms embody the notion of ordre publique and if the notion of Yuskogens is not difficult to understand it is nevertheless quite troubling in a legal order based on the sovereignty of states and resulting from their consent and indeed Yuskogens means that despite their sovereignty states are not entitled to validly consent and make treaties about whatever they wish. Yuskogens is out of the reach of states it is above them and they must always comply with it since they cannot escape it by concluding treaties which would pretend to dispose of it. Any such treaty would be null and void from the start or as stated under article 64 of the Vienna Convention it would become void in case a new parametry norm arises after the treaty has been concluded. The conceptual novelty introduced by Yuskogens and the perceived limitation to state sovereignty it entails were not easily accepted and for many years the issue of Yuskogens has been very divisive. However international courts and tribunals the European and the inter American courts of human rights the international criminal court the international criminal tribunal for the former Yugoslavia and finally the international court of justice those courts and tribunals have gradually referred to the concept of Yuskogens so that its very notion is undoubtedly part of today's international law. But what is the nature of Yuskogens and how do Yuskogens norms come to existence? Well there have been endless debates about the nature of Yuskogens. Is Yuskogens part of Kashmir international law or is it something completely different? Does it belong to the category of general principles or to another specific category of sources? In a case between Belgium and Senegal about the failure by Senegal to prosecute the former head of state of Chad accused of acts of torture the international court of justice briefly said this and I quote in the court's opinion the prohibition of torture is part of Kashmir international law and it has become a peremptory norm Yuskogens end of quote. The court went on to refer to practice and to opinion Yuris so from the ICJ's point of view it seems undisputable that Yuskogens norms are part of customary international law. They come to existence through the usual customary process. However and because each and every rule of Kashmir international law is not peremptory Yuskogens norms are a special category of norms within Kashmir international law and this is because of the specific opinion Yuris that is required for their formation. Article 53 of the Vienna Convention makes this very clear by saying this and I quote again a peremptory norm of general international law is a norm accepted and recognized by the international community of states as a whole as a norm from which no degradation is permitted and which can be modified only by a subsequent norm of general international law having the same character end of quote. This is the specific opinion Yuris. This is all good and well but which concrete norms can be said to a pertinent to Yuskogens if one takes the threshold expressed by article 53 seriously not many rules of Kashmir international law can be said to be peremptory and it is of course very tempting to resort to the language of Yuskogens in order to affirm that a rule is very important but this should not be done lightly. So for the time being Yuskogens is made of only a few new norms and the collection of norms is not extensive. There is no official exhaustive list of them but from the case law the following Yuskogens norms can be gathered the prohibition of wars of aggression the prohibition of genocide of crimes against humanity and of war crimes the prohibition of slavery the prohibition of apartheid and probably also of racial discrimination or ethnic cleansing the prohibition of torture as we called above from the Belgian Senegal case. There may be other rules a pertaining to Yuskogens and I do not pretend that this short list is exhaustive but the prohibition I mentioned certainly belong to the felt necessities of contemporary international law and it is good to take stock of the progress they represent in the common conscience of humanity. Let me make two final remarks about Yuskogens and the first remark is as follows the concept of Yuskogens was introduced in the Vienna Convention as a ground for contesting the validity of treaties but so far states have hopefully refrained from concluding treaties that would be contrary to parametry norms. So invalidity of treaties for breach of Yuskogens is rather hypothetical. However this does not mean that Yuskogens is not important as a legal concept on the contrary having parametry norms within international law profoundly changes that legal order as it conveys the idea that certain core values are common to mankind and must be protected by norms that cannot be transgressed. Moreover and as we shall see later when discussing international responsibility serious breaches of parametry norms entail specific consequences. My second remark under the Vienna Convention Yuskogens is the only content oriented ground for treaty invalidity in other words the fact that the treaty is substantially unequal that one party takes all the benefit from the treaty while the other bears all the cost of it this is not a ground for invalidating the treaty and this may sound unfair but the substantial inequality of the respective party's obligation under the treaty is not a legal ground for considering that it is null and void and this of course is without prejudice to a possible defect of consent as we have studied earlier.


The Beltrami County Courthouse is looking to get over $180,000 for a new audio and video system. During the county board meeting this week, the commissioners heard about the state of the current equipment that was installed 11 years ago. Most of the money needed from the county's capital improvement fund would be used for the audio and video equipment and installation. Because the equipment would generate lots of heat, the courthouse would also need a new AC system as well. Constantly taking this or constantly taking that to get it fixed. Here we are 11 years later and I'm saying, hey, we've got to quit nickel diming this thing and get it up for once so we don't have to look at it again. The board of commissioners decided to not take any action at this time. If you've enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland Public Television.


All right, in 1 Corinthians chapter 10, beginning at verse 1, reading to verse 4, Paul writes, moreover, brethren, I do not want you to be unaware that all our fathers were under the cloud, all passed through the sea, all were baptized into Moses in the cloud and in the sea, and all ate the same spiritual food, and all drank the same spiritual drink, for they drank of that spiritual rock that followed them and that rock was Christ.  I know that verse 4 probably was thought of as a platform where it speaks about drinking, because we have some former drunks in here, myself the foremost. And so drinking, oh yeah, let's talk about drinking Christ, but we'll look at that in just a moment. Let me share a couple of things with you and then we'll get into our study, little introductory thoughts. I got saved in 1970 when I was one year old and I got saved when I was 20 years old in 1970, and like many of you came out of a background of a variety of things, including alcohol and drug abuse and the things that go along with that. Many times when I speak into our congregation and I say the things that went along with that, there are those who can hear what I'm saying, they know exactly what I mean by that. There are others who don't have a clue what that means when I say those things that are associated with drugs and alcohol, but those of you who had that kind of bondage know the kinds of things I'm talking about, the stealing that you would do in order to buy your drugs, you know, the immorality that went along with the drinking and the drug taking, and all of those kinds of things, the violence that you can have, that because you're drunk and you think that you suddenly grow six inches and weigh 250 pounds, I mean those kinds of things. And so when I speak about the things that went along with that, that's what I'm referring to, the lifestyle, the things that you did, the things that you were. You know, we all have testimonies, don't we? If you're saved, you have a testimony and you can speak concerning it. You can say this is what I was and this is how I lived and these are the things that I did, but only you and God know your real testimony, right?  There is the testimony you give so that people will listen to and understand where you came from, but you do not say all the things you really did, you never will. Only God really knows what you really did. Only God really knows what you really thought. Only God knows what you would have done if you had the opportunity to do those things, and those are also the things that Jesus Christ set you free from. And so when I got saved, you know, the Bible, it speaks very clearly. If any man be in Christ, he's the new creation. Old things are passed away. Behold. And that word behold is a big word. Behold. What a miracle. How amazing. Behold. All things are become new. All things are become new. God has washed away all of my sin. God has washed away all of my past. He's taken my past and dropped it into the sea of his forgetfulness, never to be retrieved, never to be brought up again. As far as the east is from the west, so far has he removed my sins from me. Bless the Lord for the new life that Jesus Christ gave to you. Bless God for it. Don't ever forget it. It wasn't it wasn't it wasn't something that was cheap. Because sometimes we think, well, if it's cheap, you know, if it's free, it's cheap. No, it's free, but it is not cheap. The blood of Christ is not cheap. And when Jesus poured his blood out for me, I appreciated that. I appreciated it. They didn't understand it that deeply. How can you when you're first getting saved? How can you really understand that? And it's taken me 43 almost 44 years to grow in my appreciation for the grace of God, for the goodness of God, for the love of God, for the compassion of God, for the mercy of God, for the justice of God, for the holiness of God, for the righteousness of God. It's taken me a long time. And I'm still on that path of understanding. But I do know one thing. Once I was lost, but now I'm found. Once I was blind, but now I see. I do know that and I bless the Lord for that because God has made me a new creation in Jesus Christ. And so someone says, well, I'm a drunk, you know, but I haven't had a drink in 20 years. You know, I'm a recovering drunk. I'm not a recovering drunk. I'm a recovered drunk.  I've been covered by his blood and I identify with the work of Christ, not the past life. I identify with what Jesus Christ is doing and has done, not what I have done and have been. You need to start seeing yourself for what you are.  Don't let the world tell you that you are simply being recovered. You are recovered.  Put on the mind of Christ and understand this. God is able to do abundantly above all we could ask or think. He can transform you and he does transform you and he regard you already as glorifying him because as far as God is concerned, the work is done. When Jesus died on that cross and said, it is finished. This is a finished work we're talking about. You're in process, but in God's sight, it's a done deal. You are new in him. Understand that. Understand that today. And so I better get into the Bible study. That was just on my heart. See, when we, when we got, when we got saved, when I got saved in 1970, I was part of the Jesus Freaks, Jesus movement. It's all about Jesus, Jesus music, Jesus people. It's all about him. It never was about Calvary Chapel. Never was. It's always been about Jesus Christ. It just so happens that I go to a Calvary Chapel and pastor Calvary Chapel, but it's always been about Jesus Christ. He's my savior, not Chuck Smith. He's my savior, but I loved my pastor Chuck Smith because he was my spiritual father and he led me in the ways of the Lord. And I have a son's loyalty to a father and I love my pastor to this day and I will love him forever because he's still alive, though he's passed on to heaven, he just changed residence. He's still alive in Christ, more alive now than ever before. And I still have a love in my heart for that man, you see, but he poured some things into me through the word of God that have helped me in my pastoral and spiritual life. And some of those things maybe I'll be able to share with you today. Now, I'm going to look at this as a teacher, but I hope to be able to take some time to be what I am actually an exhorter and you'll see that in just a moment. As we look at this passage again, First Corinthians chapter 10, we need a context. And we begin in our, we begin our context by remembering chapter eight. You don't necessarily have to turn there, but you may be able to view it. In chapter eight of First Corinthians, Paul set forth the truth that Christians are now free in the Lord Jesus Christ. And he did this by contrasting three things. In chapter eight, he contrasted three things. And I'll use a word, words that start with the letter L. He, he contrasted three things, license, legalism and liberty. Those are three things that, that Paul contrasts in chapter eight. And so to understand this contrast between license, legalism and liberty, we need to remember that many of the Corinthians were formerly idolaters. These, these people had worshiped worthless idols, but they have come to a knowledge of the true God. We know that during the time of Paul, Greek society was a pagan society. And in their religious system, there were food offerings that were made to their gods. So believers began to be confused about what was permissible for them as Christians and what was forbidden to them as Christians. And that question was posed in the form of the things that had been offered to idols. The question that they had formulated, and you need to remember that in 1 Corinthians, Paul is actually answering a series of questions that he's been asked relating to the Christian faith. You need to remember that at the heart of all of this is the question, is it permissible for a Christian to eat food that has been offered to idols? That was a question that Paul had been answering. So at the heart of that question is what would be called Christian liberty. What is permissible if scripture does not specifically forbid it?  We've argued over drinking alcohol. We've argued over smoking. We've had arguments over drinking coffee, playing cards. The church has actually argued over whether women should wear makeup in jewelry. Now me, I'm in favor of women wearing makeup, but not men. The church has argued about dancing, either inside or outside of the church. The church has argued over styles of worship. You don't have a pipe organ, so you must not be worshiping Jesus Christ. You have an electric guitar, you're worshiping the devil. They've argued like that. The church has argued about going to movies or listening to secular music. The church has argued over the length of men's hair, how to dress in church. The church has argued over tattoos that's doing that today. Body piercing, you know? Body piercing. I mean, I see some kids and they come in, they have these body piercings and I look at them and I think that has to hurt. That had to hurt. How could you? Anyway, body piercing. They argue over children in worship services, the length of church services, the order of church services. They have argued, some of you know this, whether we should stand up for the entire church service, and some churches they do this day, they stand. The congregation stands the entire church service or whether we can be seated. I mean, there have been so many arguments over things that in many ways really are not the most important things. Now many feel that these are spiritual issues, but they really don't have a scripture that actually speaks to that issue. So what that has developed is for some to practice any or all of these behaviors, well, that's just sin. But for others, these issues are debatable and should not divide the church. And so throughout the history, the church has generally followed two extremes, either legalism or license. So I'll define that for you, legalism. When I speak about legalism, legalism is what has been called a fleshly attitude that conforms to a code for the purpose of exalting self. So a legalist sees every behavior, attitude or habit as either black or white. When you look into scripture, you find the Pharisees. The Pharisees were legalists. They had rules and regulations that covered every aspect of life. And so you have legalists. But then you have those who are more inclined to what would be called license.  Everything for somebody in that particular mindset, everything's white. Now when living under license, what happens is people begin to no longer care about what the Bible has to say. And what that develops, and we see that today, is Christians who live in a way that is left up basically to personal interpretation and preference. We just do what we like because we're free in Christ. And their rallying cry is grace. And anybody who doesn't agree with them and their interpretation of grace is just ignorant. I was reading something about this particular pastor who pastors a church in another state in the state and the church that he pastors has 16,000 members. That is a huge church by anybody's standards. And yet he's well-known because just recently he said this, he's well-known for saying this, and he has a sermon that went viral, and these are quotes from his sermon.  He said, I grew up in a church community. I was told I was a bad Christian if I had a beer. I know there are different franchises around the country that say if you drink beer, you don't love Jesus. And I bought that for a long time. He goes on to say, well somebody writes, perhaps his view on rules and regulations in Christianity, our best summarized in comments made at the end of the sermon, and this is his conclusion. He said, come and see Jesus.  He's enough, screw the rest. Now I have to watch myself because that bothered me. And I don't have that in my notes, and I don't have a certain amount of time to speak. No, I'm not going to say it. I'll keep my mouth shut. I'll let other people speak. That's what we're seeing today. Christianity by vote. Christianity by feeling. Scripture is interpreted by how a person feels after they hear what Scripture says. So a lack of discipleship has been the result. For them everything is relative. There are no guidelines for living. Now I want to make an important note here. It's not legalistic to be obedient to the word of God. Christians are to be Christ's disciples. And we follow his orders. As a soldier follows the general. As a servant follows the master. As the scholar follows his teacher. As a sheep follows its shepherd. Just so at the professing Christian to follow Christ. Obedience to God's word is not legalism. Because legalism produces bad fruit. Christian morality differs radically from legalism. Legalism is obedience to the letter of the law, to the neglect of the spirit of the law. Christian morality is obedience to the spirit of the law as well as the letter. Paul said he has made us competent as ministers of a new covenant. Not of the letter but of the spirit. For the letter kills but the spirit gives life. So how do you define what is permissible if you have no specific Scripture? Freedom in Jesus is balanced by love for him and love for other people. Our liberties are voluntarily subject to our love for other people. Liberty must never cause a weaker brother to sin. Liberty contains the power of choice, the freedom of doing as one pleases in the Lord. And just because I might feel freedom to do something does not mean that I should do that. You see I'll bring this in. Here's an application in illustration. I, like many of you in this room, was not a social drinker. There are people called social drinkers. I wasn't a social drinker. Social drinkers are those guys or those ladies who can have one drink and they nurse it throughout a night and they have it as part of their night. They're a social drinker. They have one beer, one glass of wine. I wasn't a social drinker. When I was 17, I started my night out by drinking a half gallon of wine and a quarter beer. That is not a social drinker. I drank with the idea of getting drunk. I didn't even like the taste. I liked being drunk. Over time I began to like certain tastes but when I began it was just to get drunk. There was a goal. So I never was from the time I began drinking like at 15 until I got saved. I never was one of those guys that would sit with you at a table while you had a beer and I had a beer and we just talked about the Dodgers or God forbid the Angels. Never the Giants. Never the Giants. If you got up and walked away from the table, finders, keepers, your drink was mine. That's the way it was. If you left your dope around me, finders, keepers. It was mine. And I say, man, you know, and that was just me. I was not a social drinker.  I never just took a hit off a joint. Just one hit, just one hit. No, I smoke until I passed out. I'd smoke until I passed out. And that was just my lifestyle. So when somebody comes and says, oh, you're free in Christ, you can have a beer, why would I return to the vomit? Why would I return to the mud that I've been washed from? Why would I use the grace of God to return to old habits that Jesus set me free from? I don't want that way of life. That's why I got saved. And when a pastor stands up and makes issues concerning beer and telling people you can drink, he doesn't know what addiction is. He doesn't have a clue. You do, don't you? You know what it's like to wake up in your own vomit? I do.  I know what it's like to be in a police station with my best friend barfing on my face all the wine we drank. We drank three bottles of this very cheap wine. I got arrested on Washington's birthday. They put me in the Norwalk Sheriff's substation. I'm passed out on the floor. My friend Bill is leaning over me, vomiting on my face while the sheriff, the deputy, is laughing, saying, hey, guys, you've got to see this. And I'm saying to my friend Bill, stop it. And Bill's saying, I can't. I know what that's like. These beer commercials where these guys are all fit and in shape drinking Miller light, you've got to be kidding me. Show him a real guy with a real beer belly just with no teeth. That's what it's all isn't it? Isn't that really what it's all about? That's really what it is. Why would I go back to that? And why would you? And why would you? Is wine better than Jesus? Is the wine better than the Spirit of God? No, it is not. Once you've tasted the new, you never go back to the old. You don't need the old anymore. I've got the new wine of the Spirit of God. That's Christianity. Anything less than that isn't Christianity. It's a pep talk. It's an inspirational saying. I preach a Christ who sets the captive free. And you can be free. You can be free. Not everything I may be free to do results in my being conformed into the image of Jesus Christ. That's true for various, various things. As a believer, I felt I could drink if I wanted to. But the fact is, I don't want to return to the vomit. And 1 Corinthians 6-12 is a powerful scripture for me. All things are lawful for me. But all things are not helpful. All things are lawful for me. But I will not be brought under the power of any. You see, we are to do all things through Christian love. That's what 1 Corinthians 8-9 says. And we're not to be a stumbling block to somebody who's weak. We're to seek to be an encouragement to them. Legalism, license, or liberty. But the liberty I have has another L. The liberty I have in Christ is tempered by love. Love for other people. Paul said, let us not judge one another anymore, but rather resolve this, not to put a stumbling block or a cause to fall in our brother's way. So with that in mind, Paul has been illustrating how our exercise of freedom in Christ affects other people. And so chapter 10 illustrates how our use of liberty affects our own lives. Now, if you were to have a verse that was the main point of chapter 10, it would be verse 12. Let him who thinks he stands take heed lest he fall. Prideful over confidence will inevitably lead you to fall. Proverbs 29-23 says, a man's pride shall bring him low, but honor shall uphold the humble in spirit. If you think you're standing, take heed. You need to be aware of the fact that the most valuable vessel in the hand of God is the broken one. Never forget where you came from. I have never forgotten where I come from. It's helped me not to judge people who are still there. It's helped me to love them. It's helped me, because you can get guys. You can get to the point where you think, how can you do that? But for the grace of God, I was that and worse. I was that and worse. So I exalt the grace of God. I rejoice in the grace of God, because he knows what I was. And I know somewhat what I was. And yet he loved me and saved me anyway. And that amazes me. The love of God amazes me. And so it should break you. It should humble you. You see, the Corinthians were in danger of believing that they were beyond stumbling. They were saved. They were baptized. They were well taught. They were spirit-filled. They were maturing in their faith. So in their eyes, they may have thought that they had been arriving at their spiritual maturity. And so they got to the point where they thought they were beyond temptation. They were now freely associating with pagans in their social activities and their social ceremonies. They thought they were beyond falling. Now, of course, we're not to withdraw ourselves completely from other people. Jesus associated with unbelievers and he loved them deeply. But he did that in order to minister God's love for them. Not to party with them. But on the other hand, we're not impervious to influence that will be exerted upon us. We need to be aware of the fact that there are things that we've placed ourselves in a situation where we're around our old friends and they're doing the things we used to do. We need to understand that we're not as strong as we think we are. And we can be drawn back. We have to be careful. Like it says in Proverbs 12, 26, a righteous man is cautious in friendship. The way of the wicked leads him astray. 1 Corinthians 1533, do not be deceived. Evil company corrupts good habits. So Paul had said he doesn't wanna be disqualified in ministry. He wanted to remain used by the Lord. And he was pointing out that pride disqualifies you from serving the Lord. So in order to develop that, Paul began to speak of ancient Israel as an illustration. And notice with me in verses one through four here in chapter 10, he uses incidents from Israel's 40 years of wandering in the wilderness. So he's saying basically, when he says, I don't want you to be unaware, he's saying don't ignore the lessons of the past. All that you have learned of the Lord should be remembered constantly. That's why we should be reading the Bible by the way, because the scripture in Romans 154 says everything that was written in the past was written to teach us. So that through endurance and the encouragement of scripture, we might have hope. And he begins now to rehearse. He rehearses scripture, rehearses the history of Israel. And he wants to communicate to them certain things. He wants them to know that all the nation of Israel experienced the grace filled deliverance from physical bondage. He wants them to know that those benefits are spiritual in nature. And he wants them to know that the benefits come from Jesus Christ. So God had done a work. And he wants them to understand that. Now he says in first, he says, all our fathers were under the cloud. All Jews were physical descendants of Abraham. But he's writing to Gentiles. And so he's making it clear that through faith in Jesus they are now Abraham's spiritual descendants. It's interesting in the first four verses he uses the word all five times. That shows oneness and experience. The nation together experienced these events. He says all were under the cloud. Meaning God delivered the entire nation. Believer as well as unbeliever. He says all passed through the sea. God brought them all to safety. He says all were baptized into Moses. All recognized Moses as their leader. All ate the spiritual food which speaks of manna. Revealing that God has the grace to provide. And finally he says all drank the same spiritual drink. They drank from that rock. That rock he says was Christ. The presence of Jesus was with them, protecting them, guiding and providing for them. The way he does for you. And that's all written to provide us with one impetus, one thing. And I'll close with this because my time is up. But let me say this, that rock was Christ. That water was Christ. That theme is gonna be built on through all the teachers and testimonies. But let me say this. We do not need the water of the world any longer. Because the water of the world only makes you thirstier. When you drink of what the world says it has to offer you, materialism, sexual pleasures, whatever it may be, you drink of it and you want more. It never quenches your thirst, it provokes it. Because once you've tasted of it, you can have it to it's full and you'll say is that all there is to this. I thought if I had this car I'd be happy. You get the car and then you say, didn't make me happy. I thought if I had this relationship that I'd be happy. You got the relationship, it didn't make you happy. I thought if I had this education, if I had this, whatever it may have been, that I'd be fulfilled and I wasn't fulfilled. And then one day God spoke from heaven and he said what you've been looking for, well if you drink of this water you will thirst again. But if you drink from the water that I give to you it'll bubble out of you, it's living water. We need the water of the spirit. Jesus made it very clear, if any man thirsts let him come to me and drink. He that believes on me as the scripture has said out of his belly shall flow rivers of living water. I came to faith in Christ and in the front room, in the city of Lehabra with some friends, I prayed that God would baptize me with the Holy Spirit. God fill me to overflowing, pour your water of life upon me. I wanna drink of you. And the Lord baptized me in his spirit. I didn't have a 10 step program, I didn't have a five step program. I had a one step program, one step to Jesus. And Jesus empowered me, one step to Jesus. And that's what you need today. You need the power of the Holy Spirit. Drink of Christ, he will fill you in the innermost being. He will fill you. Do I wanna go back to alcohol? Of course not. Do I wanna go back to the drugs? No.  Can you come and help me? I said, sure dad, he had a shopping bag. He said, I don't know what this is, but I think you will. And I opened it up, it was a kilo. A kilo, somebody had dumped a kilo in his front yard. And I opened it up and that odor of that weed just waffed it into my face. And I go, whoa, I need a twinkie. I rolled that bag up and turned it into the police. I don't need that. You know what I need and what you need? Jesus Christ. Jesus Christ.


There is a widespread assumption that if you grow the economy, jobs will follow. But the recent evidence is that you can have rates of economic growth that are far higher than the rate of employment growth in countries because the path of economic growth has been capital intensive rather than labor intensive. For instance, the country, India, the economic growth has been IT led led by the biotech companies and the pharmaceuticals. It has not been broad labor based manufacturing led. And so India is a middle income country with a labor market that looks like a developing country. Over 90% of the workforce is informal. Economic growth is simply not enough. We need jobs led economic growth, labor intensive economic growth.


Hey there, welcome to web.dev live. I'm Diyong Almer, and I work on the web developer ecosystem at Google, and I'm delighted to kick off our online event. First, though, I want to acknowledge the times we're in. We're dealing with a global pandemic that has taken a huge toll on us all. And most recently, we're witness to events which have once again surfaced the systemic racism in our society that we must do everything we can to eradicate. You know, these events have been really humbling. They're showing us how much work we have ahead of us, but they also show us the power of community. So we join you today and over the next three days in the spirit of being together and helping each other, because we were upset when we had to cancel Google I.O. And I kept thinking about an empty shoreline amphitheater on the days that some of us would have congregated. And web developers reached out sharing these same feelings, wishing we could be discussing ideas and enjoying the hallway track. Whether you're joining us from your couch, kitchen, or hammock, we hope you're safe and ready to kick Web.dev Live into gear. Now, we'll be coming to you in different time zones each day, reaching you no matter where you are on the globe. We'll be bringing you content from across our teams as well as members from the web community at large. Now, each day, you'll have Googlers on standby to answer your questions in real time. So as you're watching the session, simply head over to the live chat on web.dev.live or on YouTube and just ask away. Now, when coronavirus became global, we really felt the need to stabilize. This resulted in us pausing Chrome releases and temporarily rolling back the same site cookie changes. Now, we wanted to track Chrome usage and see what changed to make sure that we could be on top of any ecosystem changes too. You probably won't be surprised that we saw surges in usage of media APIs as video chat and streaming really soared. Now, also, some types of content saw large traffic surges such as food, commerce, entertainment, health, science, et cetera. And many developers were focusing on making sure these sites were as resilient as possible. That's when we gathered our best practices and made them available on web.dev slash COVID-19. We saw a lot of developers scramble to make changes to their websites and many created new ones. Governments had to jump on this to make sure that people had all of the critical information that was changing rapidly. I remember seeing Alex Russell tweet about one of these government sites from the state of California. We were really inspired with their work and wanted to ask them about their experience and they kindly agreed to join us. So let's welcome Aaron Hans, the engineering tech lead on the project. Thank you. Great to be here.  The Alpha.ca.gov team was formed in December of 2019 by Angelica Carrote to bring human-centered design processes to the state of California and improve their online services. We built a lot of prototypes for things like how to help people review the safety of their tap water and see if they're eligible for subsidized phone services and to prepare for wildfires. And then when the pandemic hit the state, we were asked to stand up the public response site. Got it. So when a government team has to build something like this, like how do you go about it? What are your core principles? The number one goal is to make something that works well for everybody. And the technical considerations, they're passing accessibility audits, making sure it works with keyboard navigation, with screen readers, and that it has a smooth experience on low end hardware. We use the cheapest phone we can get from the local Cricut wireless as our test device. And the non-technical considerations are the readability, what's the grade level of all the content, and are we really building something that users need and iterating based on their feedback? Got it. Now, I've been trying to picture the time pressure that you had here to get this site out. Can you tell us a little bit about how you actually built the site and how you manage the trade-offs between quality and that timeline? Sure, it's definitely an accelerated timeline. We put the site up in four days, then the governor announced a statewide lockdown and we had millions of visitors. Really happy that we chose a static site generator for that because it helped us weather the traffic smoothly. We chose 11 for the static site generator, and we augment that with web components and serverless APIs built on Node.js. Got it. We're actually fans of 11T2. We use it on web.dev and really like it. Was this kind of a new setup for the team to build a website like this, or have you been doing this for a while? I remember reading that article about how web.dev was built and being really happy that we were using some of the same tools. We started using 11T at the end of last year, just to use it for a blog on news.alpha.ca.gov. And when we used it for the COVID-19 site, it's built on content authored in the WordPress environment and then we consume it with the WordPress API and use GitHub Actions for the 11T production build. Got it. Now you've got the site out there now. I'm curious, what's next for you and the team? Next things for the team are continuing to respond to the pandemic. We're gonna be getting back to helping improve other online services and we're hiring. Check us out at news.alpha.ca.gov if you wanna help out. We're talking about 11T and the other tools that we're using. I wanted to mention that Lighthouse is an incredibly important tool for us because performance is such a paramount concern. And I love the way that it gamifies by development. You can get the rest of your teammates to challenge each other and say who can put up some more points today? We really need to get that score up. And I'm curious who's winning on the points race and I'm really impressed in how you think of performance being a key part of the accessibility story in general. Well, Aaron, it's been really inspiring to see the work that you and the team did here. Again, at an incredibly stressful time, thank you so much for coming on and sharing the story with us. Thank you. Now it's been great to see developers like Aaron focus on accessibility, resilience and performance. And we've made some announcements over the last month about a program that brings this all together under the umbrella of web vitals. To hear more, let's welcome Elizabeth, a PM on the Chrome team to explain. Thanks, Deanne. Yeah, there have been a lot of product updates and releases and I'm really excited to go over them with you. Yeah, it's been particularly busy here over the past couple of months. So it'd be great to have you get us up to speed on web vitals and what developers should be really considering here. Yeah, that's great. Let's dive in. First off, what are core web vitals? They are a set of user-centric metrics and thresholds that apply to all web pages across all industry verticals and all types of experiences on the web. They are signals to developers and business stakeholders about the basic health of your site and as such they should be measured by everybody. But okay, I jumped straight into definitions. Let's take a step back. Why did we introduce core web vitals as a thing? There are already tons of metrics, lots of guidance about how to measure your site's performance. How do core web vitals help us? Well, let's go back to our foundational goal. We want to create outlandishly phenomenal experience for all of our users. And it's not just out of the goodness of our hearts either. We know that every time we have a rage clicker on our site, we lose out on a reader, a customer or a client. Also, we want a money pug. So there is this mythical, absolutely fabulous experience that we've set our sites on creating. It seems easy until you realize that the unicorn horn requires both loading and interactivity performance measurement and the rainbow. Well, the rainbow requires an entire rum setup for each color. So there you are watching your flying unicend, unicorn doxioned, and you realize that you have this. It's gorged on a bit too much JavaScript. It doesn't respond when you're issuing it commands and that's upsetting. But it's gonna take quite a bit to get this to this. So the question is, where do you start? Well, in order to know if you've improved, we need to know what to measure. To know what to measure, we need to define our goals. So put another way, what makes a web experience shine? This is where the core dimensions of quality come in. There are foundational elements of a user experience that make a unicend shine above the competition. Content needs to load quickly. We've all been there. The longer we have to wait, the more likely we are to bail. So your pages have to load fast. Interactivity is just as important. You're clicking and nothing is happening. No fun. You don't just need content to be visible. You need it to be available for use. Lastly, we want a page to be stable and predictable. Just a few pixels moving around can make a huge difference. These core dimensions of quality reflect user-centric signals that have long been mission critical for you and your site's success. So we are closer to defining quality, but how do we measure these quality dimensions? And that's where representative metrics come in. To represent fast loading, we have largest contentful paint or LCP. It provides insight into how quickly a user is able to see the meat of what they are expecting and wanting out of a page. For responsive interactivity, we have first input delay or FID. This metric has been a critical signal for developers for some time to understand how long a page takes to respond to a user's initial input. And finally, to represent visual stability, we have cumulative layout shift, CLS. CLS measures the amount that the elements within the viewport move around during load time. Okay, so we know how to measure our core quality dimensions, and let's say my LCP is three seconds. Do I celebrate? Wait, I don't actually have any idea whether or not that's good. So I need to evaluate my performance on a spectrum for each metric, which is where the final element of Core Web Vitals comes in are thresholds. For each representative metric, we have clear goalposts around what constitutes a good experience, one that needs improvement and one that's poor. So for instance, for LCP, anything that is 2.5 seconds or less is on its way to being a unisoned. Anything between 2.5 and four seconds needs some work, and anything above four seconds is needing quite a bit of love. So to finish up our definition of what are Core Web Vitals, the initiative is a combination of three things. First is user-centric quality dimensions, then we have representative metrics of those dimensions. And finally, thresholds to help you evaluate whether or not your performance is good or not against any given metric. But there is one more piece of really important information. We need to know how many page loads need to hit the thresholds for the Core Web Vitals metrics to constitute a good experience. So say we have 100 users. If only one of them has an LCP below 2.5 seconds, do I pass Core Web Vitals? The answer is no. Core Web Vitals uses the 75th percentile value of all page views in the field to evaluate against the thresholds. In other words, if at least 75% of page views to a site meet the good threshold, the site is classified as having a good performance for that metric.  The 75th percentile is used to evaluate all three. Core Web Vitals is a holistic package of everything you need to create the foundation of a healthy site. They are valuable because they show you exactly where to start to set yourself up for success. If 75% of your users are getting fast, interactive, stable content, it's cause for celebration. But as we know, there are other dimensions of quality that are extremely important. Accessibility, security, mobile friendliness, there are a lot of dimensions that make a basic unisoned even more fabulous and are important to your site's success. So don't stop measuring these if you already are and if you aren't already, once you've optimized your Core Web Vitals, you can begin to venturing into measuring and benchmarking against other important vitals that are relevant to your business and your users. Core Web Vitals are just as the name indicates. They are core and provide you with a solid foundation upon which to further optimize. Given how important it is to quantify a user's experience accurately in order to be successful on the web, we are constantly working to find ways to better measure all quality dimensions. What this evolution has often meant in the past is a stream of new metrics, tweaks to existing metrics, and new guidance, many times at an unpredictable cadence. We know how difficult this can be when trying to set goals, align roadmaps, and get organizational buy-in. Because of this, we want to set a predictable cadence of updates to Core Web Vitals. They will be refreshed once a year around the time of Google I.O. to ensure that they reflect the latest in our learnings, and this includes adjustments to the set of metrics as well as the thresholds. Looking ahead towards 2021, we will be providing regular updates on future metric candidates, motivations, and implementation status. Okay, so this is all fine and good, but how do I get started? To know what to optimize, you have to measure first. And Core Web Vitals are now in all of your favorite developer tools, and there are more than what is listed here, including a new Web Vitals library and a bunch of ecosystem tools that have already adopted them. As you can see, Core Web Vitals are available across the board. You're able to measure them for a specific page, for your origin, locally in the lab, and from real users in the field. Remember that first input delay is only measurable in the field, so you have to have a real user clicking on your page in order to measure it, but that doesn't mean you can't use lab tools to help you improve it. Total blocking time, TBT, is a proxy lab metric for FID that allows you to debug and improve your interactivity in the lab before your users ever have to experience a bad FID. The next obvious question is, again, this is great, but where do I start? What tool should I use? I'm so glad you asked. Each tool has its own strength. For example, PSI is one of the only places you can see your lab and field data in one place, and Search Console is critical for identifying page types that need improvement. As I mentioned earlier, we're seeing so many great ecosystem players in production monitoring solutions already implementing support for Core Web Vitals, and we're really delighted. But again, you ask, you've shown me the magical Unisoned and now you've given me a palette of tools to choose from. That's amazing, but tell me what to do first. Okay, two things. First, go to PageSpeed Insights. That will give you a pulse of your Core Web Vitals performance in both the field and the lab. From crux, you'll be able to see whether or not 75% of your loads are hitting the Core Web Vitals thresholds for both your page and your origin in the field. Then you can take a look at your lab data from Lighthouse to see whether or not you are hitting the Core Web Vitals thresholds for each metric in a synthetic testing environment. This helps to guide you towards actionable opportunities to improve your page's performance. Second, check out some more in-depth talks later today that go into detail about measuring and optimizing against your Core Web Vitals. And with that, I'm gonna pass it back to Dion. Thank you so much. Great, yeah, thanks for showing us the context and all of the information across the whole slew of tools there, Elizabeth. My pleasure. One of the critical steps in modern web development with a lot of influence over your vitals is your build step. That's where your CSS modules are turned into real CSS. Your bundler analyzes your module graph and optimizations can really kick in. We wanted to go deeper here to understand the popular bundlers, how they work, what they can and cannot do and how to set them up for success. Let's welcome Serma to tell us more. Hey, Serma. Hey, Dion. So there are many best practices to follow in web development. Knowing them is one thing, but getting your build system to follow them as well is kind of another beast. So do you have anything to maybe report on that from? So there is two bits on this side of things. On the one hand, there are many developers who want to know what build tool they should learn and use for their next project. And on the other hand, there are many projects that already have a build tool set up but are looking to improve their output. To tackle both of these problems, we built Tooling Report. Tooling Report is a website that you can actually go to right now, tooling.report. We create an extensive list of best practices in web development, took what we think are the four most popular build tools and checked for each build tool if it allows you to follow that best practice. And each tool gets a point for each test that it passes. We chose to start this project with Browserify, Parcel, Rollup and Webpack. Now, Browserify might be surprising to some but the data indicates that there are still many sites out there that use Browserify and we want to help those projects improve their sites as well. Of course, we have been working with the core teams of all these tools to make sure that we not only use the tool correctly but also represent them fairly. The tests are subdivided into categories and in the overview, you can get a quick sense of which tool is excelling at what category. You can get more information on the test in the overview and learn more about it. And now this is where I think Tooling Report gets really interesting. Each test has a dedicated page where you can compare how the tool score on this specific test. There is an in-depth explanation on why this test is important and how it relates to best practices and web development. We also explain how we codified the best practice and what the expected outcome is. And finally, at the bottom, you can find an explanation for each tool and why it is passed or why it might have failed this test. If a tool is not passing a test, we will also link to bug reports on the tool's issue tracker. Many of them we have actually filed ourselves while building Tooling Report. We also link to a minimal NPM project that we use to determine the tool's behavior. This way, Tooling Report not only tells you what a tool can and cannot do, but you can also look at the configuration files and plugins to see how you can follow a best practice with this tool. This way, the site double functions as a source of documentation. The entire site is open source on GitHub and we'd love the community to help us come up with more tests and help us add more tools over time. So you can check this out now on tooling.report. Thanks for joining me, Soma. Cheers. Now we're all becoming more aware of the importance of security and privacy. Chrome believes in an open web that's respectful of user's privacy and maintains key use cases that keeps the web working for everyone. I'd love to welcome Rohan to have a chat and kind of share some of what's new here. Hey there. Thanks, Dion. My name's Rohan and I look after Web DevRel for security, privacy, payments and identity, or SPI for short. Now, while that's a cute internal name, we are part of the wider trust and safety team within Chrome. Great. So why don't we start with same site cookies and the temporary rollback that kind of kicked into gear for us when COVID kind of really started to hit globally? Can you kind of share what the latest news is there? Sure, yeah, so hopefully, as a lot of you are aware, there's an update to the cookie standard that's being adopted across Chrome, Firefox, Edge, and others to restrict cookies to first party by default, along with requiring explicitly marking cookies for third party contexts. Now, that's all configured via the same site attribute, hence same site cookies. We were rolling this out to stable Chrome, but decided to reverse this at the start of April because the COVID situation saw a huge jump in demand for online services, but also a huge shift in developers being at home without their equipment or looking after their families. We made the call that it was important to prioritize stability at that moment. Now, these changes are intended to make the Web a safer place, protecting against cross-site request forgery and trying to minimize the surface for COVID tracking. Sadly, during a crisis, when people are most vulnerable, you see these kind of scams in the tax jump too. So with the Chrome 84 stable release, which is mid-July or about two weeks from now if you're watching the stream, we are gonna start rolling this out again across all Chrome versions. Got it, so what I'm hearing here is that if you haven't tested your site yet, if you haven't made changes to kind of make sure that everything works well, now is actually the time to get going. Absolutely, so we have documentation and examples and samples out there right now for same site on web.dev as well as on chromium.org and we'll be covering implementation and debugging in our segment on day three. Okay, so we all love cookies, but I'm assuming there's gonna be a few more things that we're gonna be talking about in the kind of general view of trust and safety? I'll be honest with you, I am gonna talk about cookies a lot, but the rest of the team does have a healthier range of interests. Got it, okay, sounds good. So we're gonna cover things like, back in 2018, Specter kind of raised its head and as a web community started to really look at what can we do to help make sure that our users are secure. Are they gonna be kind of those type of aspects that we'll be covering too? For sure, yeah, so Aji's gonna be taking us through some of the new cross-origin opener and embedder policies or coop and co-app for short. So like you were saying, Specter was a vulnerability that in a super short summary meant that malicious code running in one browser process might be able to read any data associated with that process, even if it's from a different origin and that is super bad. Now, one of the mitigations for that is site isolation or putting each site into a separate process. Aji's gonna be running through how the headers allow sites to opt into that, along with a bunch of other benefits that it brings as well. Got it, got it, got it. Okay, so we've got restricting cross-site cookies and then we've got isolating sites to individual processes. We've got this interest in evolution. So I'm sensing there's kind of a bit of a theme here. Yeah, there is definitely a theme. So we've also got salmon mode on the team and they're gonna kick off our little segment to explain the link between these. And really it comes down to the web today is seeing this evolution of expectations regarding privacy. That includes users expecting more transparency and control over their online data and new regulations impacting how data can be used and collected. Now at Google, we believe in an open web that's respectful of the user's privacy whilst also maintaining a healthy ecosystem. So under the banner of the Privacy Sandbox, we're introducing a number of standards proposals that aim to support the use cases that let people make their living off creating web content. But do that in a way that better respects user privacy. We're also actively seeking feedback on these proposals. We're participating in all the open forums with W3C to discuss our proposals, as well as those submitted by other parties too. Okay, so the web's evolving and we're getting new privacy preserving APIs coming in and we're getting rid of the old cross-site data leaking APIs so they're kind of moving out. Exactly. And one of the ways I like to think about it with our team as well is that we're kind of all about the places where you create relationships on the web. So people should feel in control of their data when they browse around the web with a clear choice about what and where they share things. And when they do want to create a relationship like signing into a site or making a purchase, that should be simple, secure, and only share what's needed. Awesome, thanks so much for the brain dump on what we're thinking about here in the Realm of Trust and Safety row and I'm really excited to see the content that's coming later on the stream from the team where we can kind of go into more of a deep dive. Cool, thanks, and I'll see you around. Now the web has a great history as a content platform with its roots in hyperlink documents, but digital content has gone on richer and richer. We think the web has a great role to play here too. I'd like to invite Paul Backhouse to talk about a new content type that we're really excited about called Web Stories. Hey, Dion. Hey, Paul. So what are these web stories and why are we working on them? My team and I have been hard at work working on web stories and I'm very excited to share some updates with you. And yes, I'm talking about these kind of stories, you know, full screen, portrait, tap to advance, swipe to move on.  Then you'd be right. But these are not your standard walled garden stories. Current implementations focus on ephemerality and ultra low barrier to creation. But our bet is that the story's format works beyond the ephemerality use case and can become its own pillar in the open web media landscape. And that's because they're really cheaper to make than video and more engaging than a text article and really important, web stories are different to walled off stories in many important ways. Just like a regular web page, you own them, you host them and very important, you get the money from the ads, not the platform serving the stories. Because stories are really a visual format, my friends at Google Search and Discover are showcasing them in really cool ways, telling me that many more integrations are coming later this year. We think these can be a great net new traffic source for web creators. These stories look visually really compelling, but how hard is it actually to create them? If you want the web to be able to compete with the closed platforms out there, story creation needs to be as intuitive and fast for all content creators. Now, lots of people are working on making web stories a thing, but one of the things my own team is doing is bringing story creation to WordPress, the most used CMS in the world. In the form of a visual editor coming to you very soon, find out more about the beta at goo.gle.story editor. You'll hopefully see all the basic editing features you would expect, like smooth image and video handling, text controls, shape masking, and so on. But we're also working on some you might not expect, like this one we call text magic, running in real time against images from the Unsplash API here. Target on this feature makes it so that the editor always ensures text is readable, making dynamic decisions about the background, line height, and so on. I hope you like it as much as I do. Yeah, it looks really cool. You know, I can't wait to read some of these stories on the web. Thanks so much for sharing there, Paul. And thanks again to Paul and everyone who took the time to join me as we kick off the event today. You know, I'm really excited about the upcoming sessions, starting with the focus on how to make your website hit its vitals and discovery through search. Now, please enjoy the show. Remember, the whole team is here to chat with you on web.dev slash live and via YouTube. I will see you there today and I'll be back tomorrow morning for the day two kickoff.


Because of Gauss' lemma, the question of factoring a polynomial in d adjoin x, where d is a unique factorization domain versus f adjoin x, where f is the field of fractions of d, the question of factoring a polynomial, one of these two rings is in fact the same. We can factor a polynomial in f adjoin x, where we're allowed to divide non-zero coefficients, or we can factor a polynomial in d adjoin x, where we have primes and divisibility. Even if the polynomial is properly contained inside of f adjoin x, we can still factor out the denominators and factor with the resultant polynomial over here in d adjoin x. Therefore, it behooves us to establish a few strategies of identifying irreducible polynomials in either of these two rings. Now, one of these mini-strategies is to search for roots. Because by the factor theorem, a polynomial has a linear factor overfield if and only if it has a root. In particular, if we have a polynomial and it has a root in the field of fractions, then in the field of fractions, it'll have a linear factor. Then in d adjoin x, it'll also have a linear factor. Even if it doesn't have a root in d adjoin x, if it does have a root in the field of fractions, that still gives us something about factorization. It still gives us a factorization over here, even if the root doesn't exist. Because of that, we really are interested in finding roots, and this forces us to go back to the field side, because there might not be the root in d adjoin x here. Like for example, if you take the polynomial 2x minus 1, if you view it as an integer polynomial, then it doesn't have a root because the root should be 1 half, which 1 half belongs to q adjoin x, but not to the integers. But we still do have the linear factor. As we're trying to factor up integer polynomial, maybe one of the factors was 2 minus x. Well, if you're looking for roots, you're not going to find them among the integers, but you could find it with the field elements. Then once you find it amongst the field elements, then you get your linear factor and can bring it back to z adjoin x. For the moment b, at least in this video, we're going to focus on the field where we are going to have the root and consider finding roots. A very useful strategy for finding irreducible polynomials, the following is when your polynomial has degree 2 or 3. If your polynomial has degree 1, then it's a linear polynomial. You can't factor it, at least not over the field, you can't. All linear polynomials over a field are irreducible. But if you have, if you're quadratic or cubic, the following strategy can work. You have some polynomial f of x, which is degree 2 or degree 3. Then f is reducible if and only if it has a root. Now, clearly, if you have a root, then you are reducible. This is a consequence of the factor theorem. But this is an if and only if statement. It goes the other way around. That if it's reducible, then it has to have a root. That's because it has to have a linear factor of its reducible. By the factor theorem, if it has a root, then it has a linear factor, and that's a proper factorization. That's the easy direction. But suppose that f has a factorization, a proper factorization. So g and h are polynomials of smaller degree, but they're non-constant because otherwise, they'd be a unit. And so in that situation, if you look at g or if you look at h, then because of 2 and 3, you can only break it up in one of two ways. When it comes to polynomial degree 2, you're going to have to break it up as two linear polynomials. So actually, you get two roots in that situation. Degree 3, you're going to have to break it up as a linear polynomial and maybe a quadratic polynomial. Or that quadratic polynomial could break up into two linears. But nonetheless, there's always a linear factor. So in degree 2 or degree 3, polynomial is going to be divisible only if it has a root because it would have to have a linear factor in that case. So of course, an important example to look at here is to look at the polynomial f of x equals x squared plus 1 and view this as a rational polynomial. I mean, it's an integer polynomial, but because of the previous theorem, we want to be looking at the rationals here because what if the root was a fraction, right? That's a polynomial. That's a possibility. Before going on, let me make a comment about the so-called rational roots theorem. Imagine you have a polynomial f of x, which could be written as a to the n, x to the n, all the way down to a to the 0. And this belongs to some polynomial ring whose coefficients come from a unique factorization domain, D. If you're looking for a quote unquote rational root of this polynomial, what you mean is you're looking for roots that live inside the field of fractions. So let f be the field of fractions for this unique factorization domain, D. So I'll make a comment about D is a unique factorization domain. Then the roots of little f inside of capital F are of the form. They're going to look like some p over q. So like x equals p over q where p here divides the constant term and q divides the leading coefficient, like so. This is a common fact that students learn about in a college algebra class like Math 1050 at SUU. And so with this polynomial here, we look at the divisors of one divided by the divisors of one. This give us two possible rational roots plus or minus one. And a very quick evaluation shows that neither of those work because in both cases you're going to get two, which is not zero. And so this shows us that by the rational roots theorem, which I leave as an exercise to the viewer here, because of the rational roots theorem, the only possible roots to the polynomial x squared plus one, x squared plus one as a rational polynomial would be plus or minus one. Neither of them works. So it doesn't have a root in the rational field, which then since this is a degree two polynomial, that means this polynomial is irreducible over the rational field. Let's look at another example. Let's look at the polynomial f of x equals x to the fourth minus two x squared plus x plus one. Again, by the rational roots theorem, if there is a, because we're viewing this as a rational polynomial right now, if there is a rational root to this polynomial, it would have to again be x equals plus or minus one. So we can check very quickly that doesn't happen if you plug in one, right? One minus two plus one plus one, you end up with one, that's not zero. And if you do negative one, you're still going to get a positive one there. You're going to get a plus one there. You're going to get a minus one there, plus one. So you end up with three this time, assuming my arithmetic's not incorrect, which is still not, it's not zero. So this polynomial doesn't have any roots to it whatsoever. Now this polynomial is degree four. So there is one other important possibility here. So having no roots means that this polynomial doesn't have a linear factor, but because it's degree four, it is possible that the polynomial could be factored into two irreducible quadratics, two quadratic polynomials with rational coefficients that don't have rational roots, such as x squared plus one. And so let's consider that for a moment. Since f is a monic polynomial here, this is a nice thing about rational factorizations. If it wasn't monic, we could just factor out the leading coefficient from everything. You might get some fractions, but with respect to the rational numbers, that's no biggie whatsoever. And so in particular, if this polynomial was reducible, I could factor it as two irreducible monic quadratic polynomials. So that would look something like x squared plus ax plus b, and x squared plus cx plus d for some coefficients a, b, and cd. For which if you were to multiply this out by the distributive property, this would look like x to the fourth plus acx cubed plus b plus ac plus d x squared plus ad plus bcx plus bd. So if you just look at all the possible combinations combined like terms, that's how those things are going to combine. And this actually gives us a system of equations, right? Because when we compare coefficients, the coefficient of x cubed should be negative two. It's going to be a plus c. So we get the equation a plus c equals negative two. If we look at the coefficient of x, which is one, it's given by the formula b plus ac plus d. So we end up, I'm sorry, that was the linear term, which is ad plus bc equals, that equals one. This function doesn't have a quadratic term, right? So we have b plus ac plus d actually equals zero in that situation. And then the constant term bd, right? This should equal one right here. And so this then gives us the possibilities. But this is a great thing, even though we're working over the rational numbers for which there are a lot of possibilities here. By Gauss's lemma, we can look at the integers here. We can assume, I mean, since f is in fact an integer factorization, we can assume that these factors have to be integer polynomials as well. Gauss's lemma's awesome with this regard. Because over the rational field, there's a lot of possibilities with these equations right here. But with integers, it gets a lot more restrictive. After all, if you have a product to integers that are equal to one, that tells you that the only way you could do that is b and d are both one, b and d are both negative one, like so. In particular, b and d are the same thing, in which case they're equal to plus or minus one. But then when you look at this equation right here, all right? Actually, nope, I wanna look at this one, J.K. If you look at this one, if you plug in one for b and d, you're gonna get a plus c equals one. Or if you plug in negative one for b and c, then you'd end up with negative a minus c equals one. Or alternatively, a plus c equals negative one. So in particular, a plus c is gonna equal one or negative one.  And so we end up with a plus c is equal to plus or minus one. But wait a second, if you look at this coefficient, we have a plus c has to equal negative two. You can't do that over the integers. You can't be negative two and one or negative one. We get a contradiction here. In which case this tells us that the polynomial, let me bring it back to the screen. Our polynomial f here, in fact, is irreducible. It doesn't have a linear factor because it doesn't have a rational root, but it also doesn't have a factorization using irreducible quadratic factors. Now in many situations, this brute force approach to proving the irreducibility of a polynomial is quite possible and could be practical, right? It essentially works well for quartic polynomials because if you're a quartic polynomial degree four, if you're reducible, you have either a linear factor, which corresponds with the rational roots that we just talked about. Or it has to have these two irreducible quadratics and you could try to play this game with the nonlinear system of equations and if you get a contradiction, it means it's irreducible. If you don't get a contradiction, you probably actually found a factorization and voila, Bob's your uncle, you have your factorization. But in general, this technique is not going to do well. Like when you start getting degree five, degree six and beyond, it's going to be very difficult to brute force all the possible factorizations and so it would be nice to have some other criteria. In the next video, we'll talk about Eisenstein's criteria, which is actually a very useful technique for showing that arbitrarily large polynomials can be in fact irreducible.


Hi friends, welcome back to my channel and a very special welcome if you're new here. My name is Jen I'm a certified weight loss and nutrition coach. I follow weight watchers and I track calories and macros Happy Friday. It is Friday. So I'm gonna talk to you a little bit about my week my weigh-in And we're gonna talk about this week's ww workshop topic So if you're excited give this video a big huge thumbs up Make sure you're subscribed and your bell notification is turned on because I do a weigh-in every Friday And I do upload five videos every single week check out the description box down below where I will link nutrition coaching I do offer personalized macros and calories. This is what I followed last year and lost 90 pounds This is also what I continue to do to lose almost 140 pounds and maintain that weight loss I also offer one-on-one coaching for accountability or if you would like to chat with me directly Links and discounts to my favorite things and of course my facebook group come on over join us We would love to have you are all down in that description box So let's jump into my week my weigh-in and the weight watchers workshop topic The amazing week we had a really rough week. I We lost our dog diesel on saturday march 25th, you know diesel He's always the sweet things sitting in the background on the dog bed So it's been a really rough week for us It was hard for me to come in and film today's video Knowing that I wouldn't see him in the background in my future videos is is really devastating If you didn't know diesel suffered from quite a few medical problems and over the last couple of weeks His back leg was pretty swollen He has quite a large mass on his side that he's had for a couple of years We're pretty sure that it's some type of tumor Maybe not a cancerous tumor but some type of tumor that tumor spread and caused a blockage in his lymphatic system which caused his leg to swell and It got progressively worse over the course of a couple weeks It was getting to the point where his other back leg the foot was becoming swollen as well So the blockage was spreading. He wasn't able to get up on his own We actually had to help him get up and he had quite a few open wounds From putting all of his weight and laying on one side of his body It was very devastating and we had to make the hard decision Because we love him so much and for his quality of life that we had to let him go It was the hardest decision That we've ever had to make We had a hospice home. We had a home hospice event come to our home Diesel ate his head on my lap and we saw him cross over at the rainbow bridge and get his wings We miss him every single day But we know that we made the right decision and diesel knows how much we truly love him So it's been a rough week for our entire family And if you remember lola also was diagnosed with lymphoma cancer About a month and a half ago. Thank goodness. She's in full remission from chemo, but We don't know how long that remission is going to last. So it's been a very Very rough week for us. Sorry. I I'm trying not to be emotional, but this is my reality right now Lucky for me. I am not an emotional eater. In fact, when I'm sad or stressed out I'm not hungry So I actually had a really hard time even getting in my calories for a couple of days post the loss of diesel Luckily, my appetite is returning a little bit and I'm trying to get back to normal life doing my workouts Trying to focus on reaching my calories and my protein every day But this week has just been a week of sadness the focus prior to losing diesel was on diesel He had cheeseburgers and fries on saturday before the hospice vet came out and since his passing We've just focused on getting back to normal and having a normal life. So before I talk More about my week. I want to say thank you from the bottom of my heart to everybody who's reached out to me Whether it's in my facebook group on instagram on facebook and sent your condolences I did a little memorial for diesel through the hospice center that came out for his passing I will link that down below if you want to lie the candle for him And my heart is so full with the hundreds upon hundreds of you that have went and left such a sweet comment And lit a candle for my boy For his passing. It seriously makes chorea and I so happy and it makes us so Fulfilled how much diesel was loved by all of you.  You love him He's such a sweet. He's such a sweet dog that you've grown attached to him as well So thank you so much so so much for all of your love and support It seriously means the world to our family other than the sadness I worked on getting my exercise in this week and again just getting back to what's our new normal life after losing diesel I also really focused on making sure that I ate my protein this week because it made me feel a little more Normal, so I focused on the things that made my life feel normal I don't really have a whole lot more about my week So I want to actually jump into the workshop topic before I share with you this week's weigh-in This week's workshop topic is really really important. This is a question I get asked all the time from coaching clients or I get DM'd about or see on my youtube Comment section and that is how do I handle a small weight gain on the scale? Our expectation is that our weight loss does this it just goes down If we do what we're supposed to do our weight should go down The reality is our weight loss is this whether we make good decisions or not The expectation is definitely not the reality The reality is we're going to have weight fluctuations and this includes small weight gains or otherwise known as fluctuations on the scale To be honest, not every weight gain or fluctuation even needs a response from you weight fluctuation is totally normal Things that can impact this are sodium the foods that we're eating Exercise being sore from working out. There are so many different factors that can impact Fluctuations on the scale you need to set an action plan for when this happens because it's going to happen And several times throughout your weight loss journey try this decide what amount of weight gain will activate your plan Maybe it's three or more pounds Brainstorm simple healthy behaviors that help you lose weight in the past Maybe planning your dinners ahead of time Pre-tracking meals before eating out switching up your activity routine going to bed by 11 p.m attending your workshops and reflecting on past success Pick one or two that you're most likely to do and enjoy and then write out a detailed plan If I gain three or more pounds I will plan three weeknight dinners and commit to my next two Workshops having a weight gain or weight fluctuation threshold is really important Especially as you get into maintenance I know for me that I'm going to allow myself about a five pound window in either direction Losing five additional pounds gaining five pounds once that five pound threshold is hit. It is back to Action and making sure that it doesn't go over that five pound mark That's when we get into trouble And that's when we can really start to gain weight back that we've worked so hard losing gaining weight when you're trying to lose Weight absolutely stinks. It's okay to feel annoyed frustrated worried But don't use that in a negative way for your overall weight loss journey Remember in right put in the forefront of your mind that weight fluctuation is normal and you should expect it But you can plan ahead on how you're going to respond and of course There's no set amount of weight gain for each person It's individual like I said, I gave myself a five pound window Some people give themselves less some people give themselves more There's a difference between weight gain and fluctuation If you see the scale go up week after week after week, that's likely a weight gain Where you see the scale go up one day down the next day Maybe up one week down the next week that is considered to be a fluctuation And not something you need to take immediate action on it's really important to know The difference between weight gain and weight fluctuation If you see that weight fluctuation or even a weight gain Think back on the things that you did in the past that led to success and re implement those into your weight loss journey I also want to share some really interesting three fast facts from Weight Watchers Responding to weight gains in a hot state can cause us to overreact to overcorrect or give up Creating a plan for how to respond ahead of time when we're in a cold state can help us reverse small weight gains And number three a plan that leans into our strengths and go to healthy behaviors helps us follow through with more confidence The thing with weight gain or weight fluctuations is have an action plan in place Have a threshold that you're not going to exceed in either direction Kind of like my five pound threshold and make sure that you understand that weight fluctuation is normal Don't beat yourself up And remember you can't gain one two three pounds overnight even in a weekend or a week It is very unlikely that that is actual weight gain or fat gain It's a fluctuation and once you realize that there's a difference between weight gain and weight fluctuation That number on the scale isn't going to have the same effect on you I can fluctuate up to three pounds overnight and when I see that number on the scale I just brush it off because I know that's a fluctuation that I didn't gain three pounds of fat overnight And once you understand all of that those fluctuations and weight loss not doing this but doing this becomes a little bit easier to handle So I really like this topic and like I said I get asked about this All the time or I have coaching clients freak out when they see the scale go up at all It's normal. Take a deep breath Continue going. Don't pull back on your calories continue doing what you're doing and eventually Scientifically the scale is going to catch up. Like I said, I really like this topic And I think it's important to have those action plans in place and to understand the difference between weight gain And weight fluctuation speaking of weight gain weight fluctuation weight loss Let's go ahead and jump into this week's weigh-in. Like I mentioned, it's been a rough week So I didn't get in my calories every single day. So when I stepped on the scale I actually lost 1.2 pounds this week and that's a lot for me at the stage of weight loss that I'm in I really don't have any weight to lose. I'm looking more to kind of maintain my weight until I have plastic surgery in may I understand that I lost 1.2 pounds But I'm betting that that's going to go back up this next week now that I'm back to eating normally Getting in my workouts making sure that I'm not Into severe of a calorie deficit But sometimes life happens and losing weight can be a result of that But you also have to understand that it's likely that that number is going to change and maybe go back up a little bit When you go back to eating normal so to speak So I'll take the 1.2 weight loss. Definitely not on the best terms Definitely not something I even really intended and it'll be interesting to see what happens this next week If that number jumps back up a little bit, which I kind of expected to do and like I said, I'm completely okay with that It's more important to me to make sure that I'm nourishing my body and fueling my metabolism While I finish out this last little bit of my weight loss journey So of course I want to hear from you guys. How is your week? How is your weigh in? What are your thoughts on weight gain versus weight fluctuation? And does that number on the scale really dictate your life? And what are you going to do to make that not happen in the future? And again, thank you so much for all of your love and condolences to our family It means the world to us and if you enjoyed today's video, of course, give it a big thumbs up Subscribe if you're not I do a weigh in every Friday and like I said, I do upload five videos every single week Don't forget to check out the description box for nutrition coaching links and discounts to my favorite things And we'd love to have you join our facebook group. You can keep up with me a little bit more day to day there It's free. It's supportive. It's really a great place to be so come on over and join us Thank you for watching. Happy Friday, and I'll see you in tomorrow's grocery haul. Bye