Okay, good afternoon. So I will now have the finish with the part about drinks by finishing the section about irreducibility of polynomials and then we start with fields and field extensions. So the last thing that we proved was the Eisenstein criterion. I think you will still remember it. And now we want to give some other simpler criteria to check that polynomials are irreducible. So a few more, this time a few other methods or simple methods to check irreducibility of polynomials. So the first one is just a talk. So this is the following remark. So if you have a polynomial of degree two or three, if it is reducible, then it must have a zero. So if f in kx, now moment k is any field, is a polynomial of degree two or three, then it follows, then f is reducible, so not irreducible, if and only if f has a zero in k. That's quite obvious because if f is reducible, so obviously if f has, I mean this direction is trivial, if f has a zero in k, then you can take, you can divide it by x minus that zero. We have seen that before. And so it's reducible. So if f is reducible, then this means we can write f is equal to g times h with f and g with g and h polynomials of positive degree, so degree at least one. So as the degree of f is two or three, one of these two degrees must be one. g is equal to one or the degree of h is equal to one. And so obviously this is, and then that means, for instance, obviously it's symmetric whether it's g or h. So if h has degree one, that means that h can be written as a x plus b for a and b elements in k, and then b divided by a is a zero of k. So this is very simple. Now, it's not so obvious to see that a polynomial does not have a zero. I mean, at least the polynomial of degree two, maybe you might be able to do that, but probably polynomial of degree three might be some difficulty. But one thing you can see on the other hand, we had this exercise. So if f is a polynomial with integer coefficients is monic, we can write f equal to sum as say x to the n plus sum i equals 0 to n minus 1, say ai x to the i. So monic means precisely that the leading coefficient is 1. So then it was an exercise. I think I said that first in that case, if a in q is a zero of f, then a is actually an integer. And furthermore, we have that a divides the coefficient a zero in z. You can find this by division with rest. Well, maybe I shouldn't say, but OK. So this was an exercise. And so thus, for instance, if you have a monic, so thus if f in zx is monic of degree smaller equal to 3, well, say of degree 3, degree 2, I think you can deal with. Anyway, then f is irreducible if and only if none of the devices of the constant term, which I may still call a zero of a zero, is a zero of f. And obviously, you can directly check. Your a zero is just one integer, so it will have a small number of devices. You can just check for every one of them. If you put it into polynomial, whether you get zero or not. And if you don't, then the polynomial is irreducible. None, OK? So if we take, for instance, f equal to x to the 3, so this would also work with the Isenstein criterion, but doesn't matter, plus 2x minus 2 if I'm not mistaken. So the divisors of the constant term, what is it? 1 minus 1, 2, and minus 2. And if you put each of them into the polynomial, you don't get zero. And so it follows f is irreducible. So this is a kind of trivial way to check irreducible in some cases. Now I want to have a slightly more interesting way to get some example which we will use later. So this is the following remark. It's also simple. So let again, now we have a general field. So let f, say, the polynomial sum i equals 0 to n ai x to the i, be some polynomial with coefficients in a field k. So we give ourselves an element a in k. So we can just make a new polynomial out of f by shifting it by a. So we put, so we say that then f is irreducible over k. So in kx, if and only if, if I replace the variable of f by x plus a, if that is irreducible over k. And this is, so f of x plus a is the obvious thing. You just take this and replace x by x plus a. So this is defined to be i equals 0 to n ai x plus a to the i. Obviously, this is also a polynomial. And this is clear because if I take the map, say, I call it sigma a from kx to kx, which sends any polynomial g to g of x plus a. So the variable x is replaced by x plus a. This is, as you can easily see, an isomorphism of rings. So you can easily see that it's a homomorphism of rings. If you take the sum of two polynomials, you get the sum. And if you apply it to the product, you get the product. This is basically by definition. That's obviously a homomorphism, and it's an isomorphism because the inverse is what one can immediately imagine. You shift it by a, and you can shift it back by minus a. Inverse is sigma minus a. And so if you have an isomorphism of rings, then obviously it sends irreducible elements to irreducible elements and to reducible ones. Because if something is a product of two elements, then the image will be the product of the image element. So thus it preserves irreducibility. Thus f is irreducible, if and only if, f of x plus a. Or I could now call it sigma a of f is irreducible. I want to apply this in one example, which later we have to do with if you want to deal with it with dealing with roots of unity. So example, so let p be a prime number in z. So then we can look at the polynomial f, which I just take x to the p minus 1 plus x to the p minus 2 plus, and so on, plus 1, so plus x plus 1. So just take all the powers of x from p minus 1 to 1. There's a plus here. So I claim that this is irreducible in q of x. So it's an irreducible polynomial over q. And we want to do this by using this mark. So obviously, as you have learned either in high school or in the first year of university, you will find that x minus 1 times f is equal to x to the p minus 1, beginning of some geometric series. So now I can apply to this sigma 1 to this. So if I take x times sigma 1 of f, so sigma 1 was this map. x goes to x plus 1. This will be x plus 1 to the p minus 1, which you can write down in terms of binomial coefficients. So this is x times this. Now I divide it by x. So I get sigma 1 of f is equal to this divided by x. So this is sum i equals 1 to p. So I get binomial coefficient p choose i times x to the i minus 1. So if you take this one, this would be the sum from i equals 0 to p, p choose i times x to the i. Now we subtract the constant term and divide by x. So we get this. So I hope you remember. I mean, I'm not quite sure whether in all countries the notation for binomial coefficients is the same. Yeah. But as long as you know what it is. So p choose i. You can write this as p times p minus 1 times p minus i plus 1 divided by i factorial. So i times i minus 1 times 1. OK. So it's a standard exercise that such a thing if you take p choose i for prime number, this will, unless i is equal to 0 or p, this will always be the result of p. So it's easy to check that p divides p choose i for 1 smaller equal to i, smaller equal to p minus 1. And obviously, p squared does not divide p choose 1, which is just p. So we are in the situation. And obviously, p does not divide p choose p, which is 1. So we are precisely in the situation where the Eisenstein criterion applies. So the Eisenstein criterion gives us that sigma 1 of f is irreducible and thus this polynomial here is, I mean, f is irreducible. And this somehow means if you want to take a number which is not 1, such that x to the p is equal to 1, then this will, let me see. Yeah. I mean, if p is bigger than 1, then this will not be a rational number. If p is equal to 1, this polynomial is also irreducible, but in that case, it's of degree 1, so it still has a 0. OK, now let's, so that was as much as I wanted to say for the moment about these polynomials. So let us now start to talk about fields. And what we really will be talking about is field extensions. So we want to talk about field extensions. So I first say the field extension is just that you have two fields. So you have k is a field contained in a field L, such that k is a subfield of L. So a subfield of L just means a field, which is a subring of L. So k and L are fields. So we are interested in studying such field extensions, in particular somehow field extensions which has to do with polynomials. For instance, one question you could ask oneself, assume we have a polynomial in kx, say an irreducible polynomial in kx. And you want to find the field extension of k, such that this field extension might be called L, such that in L, the polynomial has a 0. And do you want to study whether this is possible or what one can say about such fields and so on? OK, so first now I want to give the proper definitions. And OK, I don't think I need this now. So let's talk about field extensions and the degree theorem. Later we come to the degree theorem. So first I have already kind of said in words what the field extension is, but let me see it correctly. I mean explicitly as a definition. So definition is subring k of a field L is called a subfield. If it is a subring, it's called a subfield of L. So a subfield of a field is just a subring which happens to be a field. And on the other hand, we usually want to view the other round. We want to view not k as a sub thing of L, but L as an extension of k. So in this case, we call L a field extension of k. And we write also that L over k is a field extension. OK, this is this field extension. And so if, I assume we have such a thing, so if L over k and large k over k are field extensions, we will later be interested in homomorphisms between L and k, which are the identity between L and large k, which are the identity on small k. So a homomorphism of rings and thus of fields, phi from k to L is called a k homomorphism if it's the identity on small k is equal to the identity. So if I take any element a in small k, then phi of a is equal to a. And in the same way, we have k isomorphism. So it's an isomorphism, which is the identity on k and k automorphism, which would be an automorphism of say, large k to itself, which is the identity on small k. Later we will find that these k automorphisms form a nice group, which we will have to study in order to study the field extensions. So that's one thing. Now there's one thing. If we have such a field extension, one obvious very easy remark is that if L over k is a field extension, then L is a k vector space. And then we can be interested in the dimension of this k vector space. So remark. Let L over k be a field extension. So then let's just look at the axioms that we have. So we have that L with the addition is an a b-ring group because a field with addition is an a b-ring group. And we can look for the multiplication. We can just restrict the multiplication of L times L to L to a multiplication from k times L to L. And restriction of the multiplication on L to k times L gives this multiplication times from k times L to L, which is somehow we want to see that L is a k vector space. So this would be the scalar multiplication such that a few axioms are fulfilled just because we have a field and we have a few axioms and we can just specialize them. So such that we have the following rules. We have such that we have the distributive laws. We have the distributive laws. This is just if I take, say, a plus b, which are now a and b elements in k, times an element x and L. So we have a b in k, x, y in L, such a plus is equal to ax plus bx. This is just a special case of the distributive law in L. And we have if I have a times x plus y is equal to ax plus ay. And we have the associative law. If we take a times b, a times b, a and b are elements in k, times an element x, this is the same as taking a times b times x. This is just because in L we have the associative law and we just restrict it. And finally, we know if we take 1 in k times any element x, this is equal to x. Because after all, 1 is the neutral element of L. And so if you look at the, if you try to remember what you learned in your first years of university, you find out that these are precisely the axioms for a vector space, L, a k vector space. So thus we have that L is a k vector space. And so then the dimension of L as a k vector space we will call the degree of the field extension. So let me write this down. Definition L over k, a field extension. So the degree, which is denoted very similar to what we did for rings, L over k, for groups that was called index. The degree L over k, but it's slightly different. Anyway, the degree L over k of L over k is the dimension of L as k vector space, where we, where obviously this degree can therefore also be infinite if the dimension is infinite. Over k is equal to infinity if L is not a finite dimensional vector space. So this degree, which is a number, which is possibly infinite, is an invariant which you can associate to every field extension. So let's see what we can see about this number. The first obvious, so and we call L over k a finite extension if this dimension, so this degree is finite. So not L over k, so the extension we call a finite extension if the index, so if L is a finite dimensional k vector space.  It's kind of essentially, I mean it's in some sense completely obvious because if L over k is equal to 1, then it means that this is a, that L is a one-dimensional k vector space and as a basis we can take the element 1 in L. So the elements of L are just 1 multiplied with any element of k and this is k. OK, so now we want to ask ourselves what happens to this degree if we kind of make several field extensions one after another. So we take L an extension of k and large k an extension of L. So what is the relation between the different degrees? And again this is similar to what we had for the index of subgroups. So first I just introduce one more word which will often be used. So it's just a different way of saying something again. So let say large k over k be a field extension and let L over k be a field extension with L is actually a subfield is contained in k. Then we call L an intermediate field of the field extension large k over k.  And so now we can, and so the degree theorem will tell us in this situation if you have k which contains L which contains small k, how the degrees of these things are related to each other. So it's not very, so this theorem is quite simple. Degree theorem, so let L be an intermediate field of an extension k, large k over small k. Then the degree of large k over small k is equal to the degree of large k over L multiplied with the degree of L over small k. Here obviously it might be that these field extensions are infinite. So I always say that for n in z some positive integer I have infinity times infinity is equal, infinity times n is equal to n times infinity is equal to infinity. Just that's the convention so that this holds. So if any of these numbers is infinite then I mean, if any on this side is infinite then also this is infinite and if this is infinite then one of these two must be infinite. So in particular we have that the extension k over k is a finite extension if and only if both of these are finite if and only if L. So large k over L and L over small k are both finite extensions. OK, so this is almost more difficult to state and prove. So first I'm actually not really interested very much in the case that any of these numbers is infinite so it's mostly interesting the case for finite extensions. And it's kind of obvious also that if k over L is not finite, L over k is not finite. Then it's clear k over small k is not because you find an infinite set of elements in large k which are linearly independent over L then they are also linearly independent over k. Or you find an infinite set of elements in L which are linearly independent over small k then this is also an infinite set of elements of large k which are linearly independent over small k. So in any case you find if any of these two degrees is infinite then this degree is infinite. So thus we can assume that both these numbers are finite. Maybe I should have done it with the index though because that's what I meant. So as we assume that these are both finite so let's say now I write a term that L over k small k and k over L are both finite extensions. Then we need to show that k over k is a finite extension and that the dimension is the product of these two indices, these two degrees. Well, we just do this by exhibiting a basis. So let, so we assume, so let, so the degree of L over k to be n and the degree of k over L to be m. And so we choose a basis first of this over this. So let x1 to xn be a basis of L over k. So as of L as a k vector space. And y1 to ym be a basis of k as an L vector space. And we want to show that by taking all possible products of these elements we get a basis of k over small k. If we take the set of all xi, yj, i goes from 1 to n and j goes from 1 to m is a basis of large k as a small k vector space. Well, if you want to show that something is a basis we have to show that it generates the vector space and that the elements are linear independent. So we just do it. So first I show they generate as k vector space. So I must be able to write any element in large k as a linear combination of elements of these things with coefficients in small k. So if I take, so if say y is an element in k we can write y equal say sum i equal 1 to m bj yj for some elements bj in L. This is as the yj's basis I can write any element in k as a linear combination with coefficients in L of these. Now for each of these bj's I can do the same and express them in terms of elements of small k for all j. We can write bj as sum, so this was a j, i equals 1 to n aij xi. This is again just the xi from a basis so where the aij obviously are elements in k. This is again because the xi from a basis of L over small k and so any element of L in particular bj can be written as a linear combination of them. So now we just put this into each other. So thus we have that our given element y can be written by right putting this description into this. So this is the sum of all i equal, so say i from 0 to n, sum of all j from 1 to m aij xi times yj. And so we have indeed written y as a linear combination of these products xi times yj. So they generate. And now we have to show that they are linearly independent. It's kind of basically a similar thing. It will be only easier if it just does it step by step. So they are linearly independent. So if sum aij xi times yj is equal to 0, 1. So i and j go again from 0 from 1 to n and from 1 to m. If I have such a linear combination which is 0 with elements aij in k, then we can first, we know that this yj are linearly independent. So if I have any combination of the yj's which is 0, then the coefficients of each yj must be 0. So as the yj are linearly independent over l, we find that and we have some combination of them which is 0, we find that all the coefficients are 0. It follows sum i aij xi is equal to 0 for all j. And now we apply this again. These xi are linearly independent over k. And we have a linear combination of them which is 0. We have that it follows that sum that aij is equal to 0 for all i and j. OK, so that was quite simple. In some sense, I think it's the obvious thing one would think of doing if one would be given this as an exercise. But anyway, that's it. As a corollary, I just want to mention like we did for the groups. So corollary, if k over k is a finite field extension and l is an intermediate field, then clearly we have that, I don't know which one I wanted, that say l over k divides k over k in the same way also large k over k. So anyway, that's clear because of the degree theorem. And so in particular, if k over k is a prime number, there are no intermediate fields except for large k and small k itself. So there's nothing I have to say to this. It's kind of completely obvious. So anyway, so this degree theorem is not very deep result, but it's very useful, obviously, because it gives us some way how to see whether certain field extensions can be there and to compare them. So now I want to talk about algebraic field extensions and then simple algebraic extensions. So now a field extension l over k will be called algebraic if every element in l is a 0 of a non-zero polynomial with coefficients in k.  So let me set this up. So for the moment, we fix the field extension. I mean, for the argument now, we fix the field extension k over small k. So we say an element a in the bigger field will be called algebraic over the smaller field if it's the 0 of a non-zero polynomial. So if there exists a non-zero polynomial in so which would be sum i equals 0 n a i x to the i with coefficients in the smaller field with f of a is equal to 0. So that means sum i from 0 to n a i a to the i. We should have called that b because it's a bit confusing. P i a to the i is equal to 0. And OK, so we will be interested in such algebraic elements. So otherwise, a is called transcendent if I can spell that over k. And k is called an algebraic extension of k if all the elements of k are algebraic over small k and large k are algebraic over k. So just as examples, we have to take the square root of 3. This is algebraic over q because it's a 0 of a. So if I take the square root over 3 in the real numbers, it's a 0 of the polynomial x squared minus 3. And if I take i, so the complex number i and c is algebraic say over q because it's a 0 of the polynomial x squared plus 1. And a very celebrated theorem of about 100 years ago is that, for instance, pi, the circumflex of a circle is transcendent. I think it's called transcendental, I think, is transcendental over q. This is however much too difficult for us. I mean, I just mentioned it. I think this was proven a bit more than 100 years ago. OK, so now we want to look at, I mean, here we'll be looking also at simple algebraic extensions. So we want to look at extension which are obtained by taking a field and adding to it one element, taking the smallest field which contains also this element. And so I want to introduce this first for more than one element and then for one element, and then we want to study them which are the simplest field extensions which we can imagine. So definition. So we still remember that we have some field extension k over k somehow. So let a1 to a n be some elements of the bigger field. So the extension of the smaller field generated by a1 to an. So first I say it loosely is the smallest field l which contains the field k. And so the smallest subfield l of k which contains k and the elements a1 to an. So this kind of saying the smallest subfields which contains them is just a different way of, it's just supposed to mean. It's the intersection of all subfields of l which contain these. It follows from the definition directly that the intersection of all such subfields l will again be a subfield which contains these elements. And it is then the smallest one which does it. And this is denoted, it is denoted k with round brackets a1 to an. So by definition, this is a subfield of k of large k which contains small k and these elements and every other subfield of k which contains small k and all these elements contains also this. So now let's look at the special case that we have only one element here. So we want to look at the subfield generated by just one element. And we want this to be an algebraic element. So, definition. So let a in our element in the bigger field k be algebraic over k over small k. Then this field generated by this one element is called a simple algebraic extension. So we will actually very much study these simple algebraic extensions. It will be very important for us for two reasons. The first one is that they're really simple. So they're easy to study and one can understand them. And the second reason is that in under reasonable assumptions all finite field extensions are simple algebraic extensions. So that these things are simple and they are also everything. So therefore we certainly should study them. So the last statement is not completely true but almost. So they are important because first easy to study and second almost algebraic extension, algebraic extensions. I will explain in a moment what almost means for instance. So this is the theorem of the primitive element which we will prove later. So if one makes some suitable assumptions, then every algebraic extension is a simple algebraic extension. And now we will see that one can completely understand them. So we want to describe them explicitly. So now I want to describe explicitly a simple algebraic extension generated by one element A. And it will be described explicitly in terms of the so-called minimal polynomial. So this is an irreducible polynomial with coefficients in the smaller field, which is monic. So the constant, the leading coefficient is one. And you can completely describe the extension in terms of this polynomial. So let me first introduce this polynomial. So the polynomial, so we already had this but I say it once more. A polynomial say f with coefficients in a field k is monic. It's called monic if it's leading coefficient is one. And so if f is a non-zero polynomial, then we can turn it into a monic polynomial by dividing by the leading coefficient, obviously. So then there is a unique polynomial say g in kx, which is monic, which generates the same idea in kx. That's kind of obvious if I, we know that two polynomials, I mean two elements in an integral domain generate the same ideal if they differ only by multiplying by unit. So that means two polynomials generate the same ideal if and only if they differ by multiplying by a constant. And so in this case, I can just obtain g by taking f and dividing it by its leading coefficient. And that's obviously unique. So now let us take an element a in our bigger field k, which is algebraic over the smaller field. And we can look at the evaluation homomorphism at a. So the evaluation homomorphism was I take a coefficient, a polynomial with coefficients in the smaller field k.  Every polynomial f is sent to the evaluation of the polynomial at a. We know that the evaluation is a ring homomorphism. So this is a ring homomorphism. So we have assumed that a is algebraic. So it means there's some polynomial, some non-zero polynomial, such that if I evaluate it at a, I get zero. So the kernel of this homomorphism is non-zero, kernel. And this kernel is an ideal. And so as this kx is, so this is an ideal. So skx is a principal ideal domain. We know that this ideal is principal. So it's the ideal generated by unique element. So there is a unique polynomial, a unique non-zero polynomial, such that it's generated by this. And if there's unique non-zero polynomial, there's unique monic polynomial, we record fA, such that this kernel of the evaluation morphism is equal to the ideal generated by fA. And fA is called the minimal polynomial of a over k. Okay, so this is this definition of the minimal polynomial. It's a bit maybe roundabout. So therefore, we immediately will give another characterization, which is easier to remember and often to use. But anyway, so if we take the, we have such an algebraic element over k, then the evaluation at a has a kernel, an ideal which is generated by unique monic polynomial, and this is the minimal polynomial. And now let me give another characterization, which is what I just said before, that it's a unique irreducible monic polynomial with coefficients in small k, which vanishes there. Proposition, so let, we are in the same situation. We have an element in the bigger field, which is algebraic for the smaller field. So the minimal polynomial, polynomial fA of a over k, is the unique irreducible monic polynomial, so f with coefficients in the smaller field, such which vanishes at a. So the minimal polynomial is, if I have an element, which is algebraic over some k, then the minimal polynomial is a polynomial with coefficients in this smaller field, which is irreducible and monic, and which vanishes at this element. And there is a unique such element, which is the same as what I defined here. This is the claim. The proof is not particularly difficult. I have to see that. So the first thing, so we take as fA the thing that we have defined here. So in order to show, we have to show that these two things are the same. So for this, I have to show that if I have this element, then this is irreducible. So let fA be the minimal polynomial of a. So we have to show fA is irreducible. Then we have found that this is an irreducible monic polynomial. And then on the other hand, we have to show if we have an irreducible monic polynomial with f of A equal to 0, we will have to show that it is this generator of this idea. So let's first show this thing is irreducible. So assume fA is a product of two polynomials. You have to show that one of them has to be constant. So we have 0. So fA lies in the kernel of this evaluation homomorphism. So fA of A is equal to 0. So if we have that it's this product, we get this. So here we are in some field. So if the product of two things is 0, then one of them must be 0. Obviously it doesn't matter which one we call g and h. So we can assume that g of A is equal to 0. I hope you remember the definition of the minimal polynomial. So say equal to 0. So then it means that g of A lies in the kernel of this evaluation homomorphism. So g lies, thus, g is in the kernel of the evaluation at A, which was generated by fA. So it means that g is equal to L times fA for some polynomial L in kx, no? But now we have fA is equal to g times h. And g is equal to L times fA. So we have that fA is equal to h times L times fA. So that means h times L is equal to 1. And in particular, it means that this h was a unit. So h is a non-zero element in k. So k star, I don't know whether I introduced it. It's just a unit. So this shows that fA was irreducible. So conversely, in order to show this result, we have to show that if we have an irreducible monic polynomial which vanishes, it's a generator of this idea of the kernel of the evaluation. So conversely, let say g be a monic irreducible polynomial, I mean in kx, with g of A is equal to 0. Then we have to see that g is equal to our fA. So it's the unique monic generator of the kernel of the evaluation map. Well, it's quite simple. g of A is equal to 0. So it follows that g is in the kernel of the evaluation, which is the ideal generated by fA. So there exists again, say, an L in kx, such that g is equal to L times fA. But g was supposed to be irreducible. If g is irreducible, it cannot be written in a non-trivial way as a product of these things unless this is a constant. So it follows, so as g is irreducible, L is an element in k star. And now, however, both g and fA are monic. So the leading coefficient is 1. So if you want to find an element in k star so that you multiply 1 by it to get 1, the only possibility is 1. So fA, monic. OK, so this is this characterization of the minimal polynomial. How much time do I actually have? I'm a bit slower than I thought. Actually, consider is slower. So as an example, so let p be a prime number. We take some positive integer. Then if you take the polynomial x to the n minus p, this is the minimal polynomial of the nth root of p over q, say, because this polynomial is irreducible by the Eisenstein criterion. And so it is, and it's monic, and so it's the minimal polynomial. So now I have not so much time, so which page is this? I'm actually not, oh, it's not so bad. Anyway, so now I will not be able to prove this theorem completely, but at least I can state it and maybe see to start with a proof. So this is the description of the simple algebraic extensions. So it's the formula for the description of simple algebraic extension in terms of the minimal polynomial. So the theorem is as follows. So we have, again, element a k, algebraic over k. And so with minimal polynomial f a, and we maybe fix the degree of the minimal polynomial. Then we have the following. First, then first, we have that k a. So the simple algebraic extension generated by a is isomorphic to k x divided the ideal generated by the minimal polynomial. Something that we have kind of seen before. And the second statement is the degree of the field extension is equal to the degree of this polynomial.  OK, what can I do with this now? So this is the statement. So one should note that this, in a suitable sense, completely describes this field. We know what the degree of the field extension is. And we know precisely what the ring structure is. We have this polynomial ring divided by this ideal. And we can also, OK, I don't have this. So maybe I can prove the first part and then go on the next time. So we have this evaluation homomorphism from k x to k of a. So it goes first to k, but the image is obviously of a because it's all expressed in terms of a, which sends g to g of a for any polynomial g. So this is a ring homomorphism. So it's a ring homomorphism whose kernel is the ideal generated by f a. We know that the kernel of the variation of a is the ideal generated by the minimum polynomial. So let now say l be the image of this map. So this is the image applied in relation of a to the whole of kx. This will be a suffering of k. So a suffering of k. So by definition, so the kernel is this and this is the image. So we find that this l is isomorphic to kx divided by this ideal. By the homomorphism theorem, we have that l is isomorphic as a ring to kx, modulo, this ideal generated by f a. So this is what we have on this side. So we will want to prove that l is equal to k of a. And also that it's a field, obviously. So as f a is irreducible, we have seen that it generates a maximal ideal in the principal ideal domain kx. So it follows that this l is a field. And if I identify the image of the constants under this map with k, because the map on the constant has, there's no kernel if I look at the constants. No, it's injective. So which contains k is a subfield. No, that's the image of the constant. Contains something isomorphic to kx, but doesn't matter. And obviously, so it contains k is a subfield. And obviously, we also have that a, which is equal to the polynomial x applied to a, is in this image, which I could call is the evaluation at a of the polynomial x is in l. So that means l is a field which contains k and a. And so thus l contains k and a. And also, by definition, l is contained in k of a, because all the elements in l are just polynomials in a. You apply a polynomial to a, so this gives you an expression of a with coefficients in k. So you stay in a. And l is contained in k of a, but thus l is k of a. If you take the evaluation morphism at a, it means you have some kind of linear combination with coefficients in k of powers of a. This certainly lies in the subfield generated by k and a. Because you have just used field operations to elements of k and a. So certainly l is contained in k of a, and this does this. And so l is equal to k of a. So this proves one. And then we will next time do number two, which is actually maybe, well, it's not more difficult. And maybe also review the statement again. I think we see each other on Wednesday or something. Anyway, so thank you.


I found one particular tactic that Trudeau, his government and his finance minister who announced it, used was particularly nasty and particularly evil and particularly bad in terms of setting a precedent. Because of the Emergency Powers Act that I guess Trudeau declared or initiated, what they were allowed to do was to basically freeze the assets of people who participate in the protest, people who funded and provided funding for the protest. Now it's true that these actions are illegal and technically you're not allowed to fund illegal actions, but all of this freezing somebody's bank account is something really, really, really serious. This is where we all have, and we'll talk about this more in a minute, but this is where we all have our money, our savings, the money we need to buy food to put on the table, almost all of us are completely dependent on the money in the banking up being available in order for us to live, to survive, to have a life. And in order for the government to take an asset like that, any asset, but certainly an asset of importance like that, shouldn't this be some due process? Isn't incumbent upon the government to prove our guilt? Isn't there something about the presumption of innocence? Now it's always the case that emergency provisions, any law that's called emergency law is scary and authoritarian and disruptive and bad and anti-freedom everywhere and anywhere. And that's true of this in the United States and the COVID national emergency order, that was originally passed, I think, by Trump and now extended by Biden, or whether it's in Canada's emergency order, whether it's all the emergency orders passed in Hungary that gave or given almost dictatorial-like powers. Emergency orders are bad, and this is part of that emergency order, but there's something particularly evil and bad about the ability to take our money with very little recourse and certainly no immediate recourse. We see the same thing. We see the same, and I would say this, if this was just the emergency act in Canada, if this was just a one-off phenomena, you know, yeah, they declared an emergency, they did this and that's it, then okay, it's bad, it's evil, it's wrong. But this is an expression of something that is going on all over the world, going on in the United States in particular, that almost nobody talks about, that is expanding in its use, that is the confiscation of our wealth directly by freezing our bank accounts, by taking our money with no due process, with no thought about the issue of innocent before proven guilty. So it seemed like a broader topic for me, and of course, you know, you can talk about, we can talk about crypto as potentially being a solution here, but the challenge is with that as well. So think about, think about in the U.S., think about the fact that we have the police, civil forfeiture laws, the police can stop you, if they suspect that you've done something wrong, they can seize what they find in the car that they've stopped, if you have to have a suitcase full of cash, they can seize it and take it, they can take your car, they can seize it, do they need a warrant, no, do they need to prove that you've committed a crime, no, I can't think of anything more authoritarian than that, the policeman feels like you might have done something wrong and why the hell should you be traveling with a lot of cash and your car must have been used in some crime or another, and they just take it. And what's interesting in the U.S. is that part of their salary, part of their compensation of the police in certain localities is based on how much they bring in through the civil forfeiture laws. The certain counties in the U.S. that you do not want to be driving in, particularly if you have cash in the car, somebody's asking if the question's today specifically about the topic. No, you generally can ask questions about anything you want, anything you want. So the whole idea of constitutional protection, the whole idea of not being able to penalize you until you've been shown to be guilty, the whole idea of due process out the window, it doesn't exist in the U.S. and I have no idea, I don't know the legal history here, I have no idea how this has withstood the court system. I have no idea how nobody is overruled us, but that's not the only law. There was an act that was passed, I believe, in the 1970s in the United States called the RICO Act. Now the RICO Act was passed in order to go after organized crime and what would happen when they'd arrest somebody in organized crime and by the time they prosecuted, by the time they found him guilty, by the time the fine was imposed on this person, all the assets that the person would have had would have somehow disappeared and therefore he couldn't pay the fine or he couldn't pay recompense to the people he stole the money from or whatever, the money had just gone. So with the idea of fighting organized crime, Congress passed a bill called the RICO Act and the RICO Act allows the authorities to freeze your assets, to confiscate your assets once you're arrested for certain crimes.  No, as soon as you're arrested they freeze your bank accounts, take your stuff and then later if you're found innocent, supposedly they give it back to you. But a trial like that can take years and during that period you have no access to bank accounts and everything else. And the idea was, well, we need extraordinary measures, emergency acts to deal with extraordinary crimes, extraordinary organizations like organized crime like the Mafia and most Americans go, yeah, okay, it's an emergency, you know, people forget the extent of which organized crime was prevalent in the 1970s and how violent America was during that period. So people accepted it. It was going to be applied to Don Corleone and not to me, not to you, just to those guys and the bad guys. So when we know they're bad guys, even before we try them, we know they're guilty before we even try them, right? So we have the RICO Act that we can take their money, freeze their accounts, destroy their lives. Before we ever, ever find out that they're guilty, that wasn't the end of it. And here I'm going to tell you one of the many reasons why I despise, despise Rudy Giuliani more than probably any other politicians out there. I despise this man because in the 1980s, as the district attorney for the Southern District of New York, the district attorney responsible for Southern District of New York, which means Wall Street, yeah, more than Trump, I despise him more than Trump. And the two of them together make a wonderful couple, wonderful couple. In the early 1980s, as the district attorney for the Southern District of New York, Rudy Giuliani started using the RICO Act for the first time on non-mafia cases, on cases that involved financiers and businessmen. He famously would arrest financiers in Wall Street, put them in handcuffs, tip the press off in advance so the press was there when he made their arrest, drag them out into the street in handcuffs, then freeze their assets, freeze their accounts, use the RICO Act, claim that they had conspired and whatever the technical legal terminology is to use it. Now these trials sometimes took years. Many of them were appealed, many, many, many, many of the cases that Giuliani tried, landed up being thrown out, landed up being reversed on appeal. Maybe five years later Giuliani wasn't even there anymore, and yet for five years, people's lives have been destroyed, their reputations in fragments, in tatters, and their assets frozen, no access to money. Now this to me was unbelievable, unbelievable abuse of power, unbelievable abuse of power. Now the law is a bad law. It should have never passed Congress. It's a massive violation of rights. But then it was passed in a particular context and it was understood by everybody that this was the context and then Giuliani blew that up. He opened it up to this day. They use RICO Act to go after business leaders and businessmen and all kinds of people. Destroying people's lives, destroying their occupation, destroying their ability to make a living. And he's a good people, productive people, people who ultimately are found to be innocent. And yet it doesn't matter. Their lives are gone, finished, destroyed. Now Rudolf Giuliani did that. He put in prison, he tried and succeeded I think in destroying the lives of many people, many good people, many productive people, many successful people. And he's never paid the price for it. He never paid the price for it.  But beyond that, he's become, after 9-11, he became a hero, a hero. So now you know a little bit more about why I despise Rudolf Giuliani.  That's something you can't live down. You're doing the lives of productive people, doing it on purpose, not caring whether they're guilty and innocent ultimately. And doing it for one purpose and one purpose only, which was Rudolf Giuliani's motivation. One purpose and one purpose only, which was to gain political power. So you can see that this idea of freezing people's accounts, taking their money, it's not unusual. Now think about, think about the banking system. Think about banks. Think about why we have banks and what we use banks for. And what assumptions do we make about our banking relationships? You know, we, you know, before there were banks, you would have to have your money on you or bury it in the backyard or under the mattress. Usually it would be in gold coins or something like that. It would be unbelievably difficult to keep it, preserve it, protect it from thieves, protect it from loss. But money that you had in those days, in that reality, the money that you had was yours. Nobody had any business in, really in how you got it. Nobody knew how much you had. There was no paper trail. There was no way to figure out, you know, maybe some transactions were documented but many were not. And you didn't have your money in one place and it wasn't concentrated and you could have, who knows how much buried in the backyard. What you did with the money was completely private. You could have invested it. Maybe they'd be a document contract to prove that. But you could buy anything you wanted to buy. Your transactions were free. You in a sense were free to transact, not free to transact because of political freedom because there was none. You were free to transact because you had privacy. Because nobody knew what you had and nobody could follow you around and tell what you did with that money. But starting at some point, in the 16th, 17th, 18th centuries, people started depositing the money that they had, merchants, people who had a little bit of money with banks or the equivalent of banks, and getting pieces of paper in exchange. They would deposit the gold in the bank. They would get pieces of paper that could be exchanged back into that gold, and these people's paper became the way in which we buy and sell things. And still, pretty much everything was private. Now, there was one aspect of this that was not, and that is that all your money was being held at a bank, at an institution that had a record of, Joe has $500 here and Sam has $50. And Joe took out $50 the other day, and again, it wasn't so much in dollars, it was in gold coins and things like that, or so and so deposited this much and took out this much. So now, while we couldn't tell what you were doing with the money, what you were buying with it, what you were investing in, somebody who could get access to the records of the bank could actually tell how much money you had, could tell what was going in and out of your account, maybe they couldn't tell exactly to whom, but they could tell a lot more about your financial life than they could when all we had was gold coins. Now, speed ahead to today. Today, we have money in a bank. The money goes in, money goes out. It's not backed by gold anymore, it's just pieces of paper or really just electrons. Every one of those electrons is coded and anybody along the chain knows exactly where it came from and who it belongs to and where it's going. If somebody seizes my bank accounts, they know a lot about me. They have access to a lot of what I do. But more than that. It used to be that we would go to the bank and take out cash and then go and shop with cash and then deposit what excess cash we had in the bank after a day of business or after we sold our stuff or whatever. And so the bank was an aggregator, but there was no record of everything we did with the cash that we had. Well, today we barely have any cash. Most of our transactions are electronic, whether wire transfers, ACH, or even checks, credit cards, PayPal, Venmo, each one of which leaves an entire trail of not knowing how much money you spent, when you spent it. But where you spent it, with whom, who you paid, who paid you. There's an entire trail now of every single dollar. What you made, what you spent, where you spent it, on whom you spent it. If you transfer money and Venmo to a friend, theoretically, the government could get records of that and could identify all of them. We live in a world in which all of our lives are completely available. Completely available. To anybody who has the ability to access our bank relationships, get our credit card information, get our bank accounts, get our 401ks, our savings accounts, everything is electronic. And if the government can access all these, we have zero privacy. Now, yes, some of us still take out cash. We still have a little bit of cash in our wallets. We spend on stuff that's cash. But even that is slowly being diminished. I bet you nobody here remembers the days when there were $500 bills because the US doesn't have $500 bills anymore. Because they don't want you to transact in high-denomination bills because they don't want you to use cash because they can't track it. The European Union, the European Central Bank, I think used to have 500 euro notes and those are gone. Not because there's no demand for those notes, but because they don't want you to. They want to make sure that when you pay for something, you pay with a credit card, with a check, with something that can be tracked, that can be followed, that can be identified. Cash is anonymous. Indeed, one could argue that there is today what some people call a war on cash. The economists who advocate for doing away with cash, the politicians, the people at the Fed, who would like to completely do away with cash. The only country where you can still get a high-denomination bill is Switzerland, where you can get a 100 Swiss franc bill still to this day. It's a little over $100. I don't know who is on the $500 bill, no idea. So this so-called war on cash takes four, what is it, forms, fuel. One is to abolish high-denomination bank notes to make it very, very, very difficult for us to do significant transactions. So they limit the denomination on the bills. Second, many countries, primarily in Europe, have legal maximums for cash payments. So this I found pretty amazing. I didn't realize this existed. 12 of the 28 European Union member states have restrictions on cash payments. For example, let's see, in Italy, you're not allowed to pay cash for anything for more than 2,999.99 euros. Like it's illegal to go buy a car and give somebody $10,000 in cash. It's illegal. They don't allow it. Why? Why don't allow it? Because only crooks pay with cash because cash is a way for you to escape income taxes because the state, the state wants to monitor what you do with your money. The state wants to know how much you make. The state wants to know how much you spend. The state wants to make sure you're paying your, quote, fair share of taxes. So the state doesn't allow you to transact. Germany's tried to put a $5,000 euro limit, but it didn't go well with the public. With the public. But France and Spain have a limit. It's 1,000 euro. Supposedly in Greece, the limit is 500 euro. I'm going to have to ask Nikos if that's true. You can't pay for something more than $500 in cash. No, that's just bizarre. In the United States, of course, doesn't have a limit. You can pay whatever you want in cash, but if you pay a business more than $10,000 in cash, that business has 15 days in which to file form 8300, the IRS financial and the financial, whatever, authorities, 15 days, $10,000. If you deposit $10,000 or more in cash in the bank, the bank has to file with the authorities that you invested. If you come into the country with more than $10,000 in cash, you have to declare it and they will file. And if you don't declare it and they find it, they can confiscate it. In other words, the goal here is to prevent you from having any financial privacy, any financial privacy and cash provides for privacy. So first, there's a Bolshevik denomination bank code. Second is place the maximum legal value in cash payments. Third, require declarations of any party. Carrying a cash amount above a specific value across national borders, we just talked about that. Fourth, require banks to report to authorities any cash deposits or withdrawals in amounts above or suspiciously near a specified value. Let's say you want to deposit $20,000 cash into your account and you're worried that if you do it all at once, they'll have to report it and you don't want them to report it. So you actually deposited in three payments, 8,000, 8,000 and 4,000. That is illegal. You can go get a massive fine for that. You can get prosecuted for that. In the United States of America, our free country, your own money, not allowed to do it. Cash is discouraged, disincentivized or taxed. Now it might be silly to own cash during inflationary times and maybe inflation is the ultimate way in which government prevents us from using cash. Because with cash, you lose value because of inflation. But think about Europe, where for the longest time interest rates would be negative. Negative. You had to pay the bank to keep your money. Pay the bank to keep your money. Well in a negative zero environment, why not have cash? But they don't want you to. They want to be able to tax you. The negative interest rate is a form of tax. They want to be able to keep the banks afloat by the banks taking a fee from you. We live in a scary world. We live in a world in which our bank transactions are all accessible to the government. It doesn't take much for the government to be able to reach in. I'll give you one other example. Venmo, PayPal, Zell. You all use Venmo, PayPal, Zell. I certainly use Venmo. I love Venmo. It's easy. I move cash like that up to a certain amount. You can't do too much. PayPal. PayPal is great. Obviously, I get some of your contributions on PayPal. Zell. Pretty cool. But you know that Venmo, PayPal, and Zell have to report any transaction over $600 to the IRS means who paid and who received it. So just know that any time you're making a substantial payment on Venmo, PayPal, or Zell, the government knows about it. The government has a record of it. No, mostly we don't care. But you never know. And of course, if they have a record of it, if they have access like that to PayPal, Zell, and Venmo, they can shut it down as well. They can freeze your ability to use those services. No, we're not capitalist guys.  We're far, far from capitalist. We're far, far from free. We're far, far from free. And you saw when the Canadian government didn't like the politics of the protesters, they would have never done this to Black Lives Matter, right? Never done this to Black Lives Matter. When they didn't like the politics of the protesters, what did they do? They froze their accounts. They made it impossible for these people to live, to trade, to transact. Well, humans don't examine this data, probably. But the data is available for humans to examine. So God, if you get audited by the IRS, they will download all their data and ask you, where did you get so much money that you're moving money in this way or that way or whatever? Where did it come from? What's the source? What's the origin? Did you pay taxes on it? So sure, they don't have somebody there looking through the pieces of paper and checking everything, but they have the record and when they need the record, they will use the record. And again, think about the fact that all these people, whether you agree with the truckers or disagree with the truckers, is irrelevant here. Think about the injustice. Think about the violation of rights of these people, money basically being inaccessible to them in a world in which how do you survive without a bank account? How do you survive without a bank account? How do you survive without credit cards? What are the financial consequences of all this? Two individuals, two living, breathing individuals. So we live in a world in which, and this is not just the US and not just Europe and not just Canada, this is global. You know, Chinese certainly have access to all these things. We live in a world in which there's no financial privacy. In a world where our financial transactions are easily accessible, a world in which the legal system is amenable to great violations of rights, the confiscation of wealth, the freezing of accounts, to really painful remedies. We live in a world in which the authoritarians can flip a switch and suddenly the money we thought was ours, the money we thought was in a bank account. In its house, I can use it. I can go buy stuff. I can live off of it. It's not mine anymore. It's gone. It's disappeared. It's gone. Now, the advantage of having cash, even though it loses value every year, the advantage of having cash is it's hard for them to get access to it. But at least you have something. Imagine an authoritarian coming to power in the United States and saying, I don't like objectivists. That's your run, Brooke.  That's one thing. I can't talk anymore. Okay, that's bad, really bad. But imagine if they also say we're taking away everything in your bank accounts. What do I do? How do I live? How do I cash out? You're finished. How do you survive? Unless you have some cash, cash. Now, this is one reason if you believe the world is going to hell, at some point in our future, if you believe authoritarianism is on the rise, if you believe there's a real risk of dictatorship, or real risk of authoritarianism, or real risk of the government penalizing its political enemies, this is why not keeping all your money in the bank makes sense. Thank you for listening or watching The Iran Book Show. If you'd like to support the show, we make it as easy as possible for you to trade with me. You get value from listening, you get value from watching. Show your appreciation. You can do that by going to iranbookshow.com slash support by going to Patreon, subscribe star locals, and just making an appropriate contribution on any one of those, any one of those channels. Also, if you'd like to see The Iran Book Show grow, please consider sharing our content, and of course, subscribe, press that little bell button right down there on YouTube, so that you get an announcement when we go live. And for those of you who are already subscribers and those of you who are already supporters of the show, thank you. I very much appreciate it.


Item Number SCP-4419 Object Class Keter Special Containment Procedures Due to the expansive range of circumstances in which SCP-4419 manifestations occur, containment efforts are to focus on information control and post-manifestation cleanup rather than physical imprisonment. Many witnesses to an SCP-4419 manifestation are to be dosed with a Class B or Class A amnestic as appropriate for their level of exposure. In cases where it is possible to restore the bodies of SCP-4419 victims to roughly standard human configurations, a cover story is to be established to explain any remaining damage as a result of their original injuries. In cases where this is not feasible, victims of SCP-4419 are to be brought into custody and, if possible, euthanized. A cover story is then to be established regarding the death of the victim due to their original injuries or conditions. Description SCP-4419 is a vehicle resembling an ambulance of Barry and Megan model, which will spontaneously appear in an area shortly before a medical emergency rises. The means by which SCP-4419 predicts these situations is currently unknown. Although the appearance of SCP-4419 differs from manifestation to manifestation, it will always resemble an ambulance appropriate for the local culture. Upon the occurrence of the medical emergency, SCP-4419 will proceed directly to the injured individual, hereafter referred to as the victim. Two individuals of Barry and physical appearances in paramedic uniforms will then exit from the back of SCP-4419, secure the victim and bring them back with them into SCP-4419. The individuals that emerge from SCP-4419 will behave as expected for a medical professional in the situation, but will repel any attempts by others to prevent them from securing the victim via extreme physical force. Once the victim has been secured within SCP-4419, it will leave the area at extreme speeds, disappearing the moment it is outside of observation. Two to seven days later, the victim will be returned outside a local area, suffering from extreme and invasive bodily modifications. Although the majority of these alterations would logically result in the death of the victim, deaths will not occur in these cases unless the modifications are tampered with or otherwise undone. The Pacific nature of these modifications differs from case to case, although there does appear to be a level of correlation with the original medical emergency. See Encounter Log 4419-1 Encounter Log 4419-1 The following is a log of encounters with SCP-4419. The original medical emergencies in each area and the bodily modifications apply to the victim. Note that this log does not encapsulate all known SCP-4419 victims, and a full record is available upon request from the Data Archive at Site-31. Date Medical Emergency Bodily Modification February 7th, 1983 A breaking car hits a pedestrian crossing the street, resulting in a broken leg. Victim returned with all limbs amputated and relocated to protrude directly from his torso. Limbs were re-amputated and a cover story was established to explain the loss of limbs as a result of a much more severe car accident. November 23rd, 1994 A man suffers from a broken jaw following a fight outside a bar. Victim returned with his jaw forced permanently open. In addition, a glass window was installed in the mouth to permit viewing of the heart, which had been relocated to the back of the throat. Due to the relocation of the heart, reversion of the body modifications was not possible, and the victim was euthanized. June 19th, 1999 A homeless man suffers from a drug overdose. Victim returned to the same place he was originally taken from.   Note that at no point was the victim's brain actually connected to his nervous system. January 29th, 2003 A wife and husband involved in the car crash suffered numerous broken bones and severe bleeding. Victims returned fused together by their backs. All bones that had been broken in the original accident were meticulously removed, resulting in the loss of use of some limbs. The victims were successfully separated, administered amnestics as appropriate, and their modified limbs were amputated. February 15th, 2006 An elderly man suffers from a heart attack. Victim was returned with 11 additional and non-functional hearts within his body stuffed between his existing organs. Exploration occurred when surgeons attempted to remove these additional hearts before agents could arrive on the scene. Surgeons and medical staff who had treated the victim were administered amnestics and the body confiscated. September 19th, 2008 A fire at a bar results in 19 people suffering from severe burns. An additional seven people suffer from skull fractures and broken bones when they attempt to confront the individuals who emerged from SCP-4419 and are also taken as victims. It is believed existing injuries were exasperated and new injuries caused by attempting to force 26 people into the limited space within SCP-4419. Victims return to the local community center as a single watery mass, which twitches and shivers when physical contact is applied. As no method of euthanizing the victims could be found, they are currently stored in a liquid tank at Site-31. November 24th, 2014 A US military private is shot while on patrol in Afghanistan due to the suspicious nature of SCP-4419's arrival and the forceful securing of the private. Other soldiers on the patrol fire upon SCP-4419's leaves. Witnesses report seeing a viscous black fluid leaking from the resulting bullet holes on SCP-4419's surface. Victim is found broken down to a thin paste and spread over the walls of their barracks the next day. The agents who initially secured the remains reported seeing a mostly intact eyeball dilate when they approached. Although what has been brought into storage is referred to for convenience's sake as remains, it is currently not known whether or not the victim has expired.


It's raining plastic. Microplastics have been found in our blood, our lungs, and now it's falling from the sky. And the worst part? Plastic never goes away. It just gets broken down into smaller pieces, blowing all around the world in a never-ending cycle. It's so bad that microplastics from Europe are landing in pure habitats like the Arctic. Even deep-sea ecosystems aren't safe. And the numbers are shocking. Researchers found that 98% of air and rainwater samples they collected from 11 national parks all had microplastic particles. So it may be invisible, but it's definitely all around us.