 This talk is the first introductory talk of an online course on Galois theory. So what I'll be doing is just giving a sort of informal overview of what Galois theory is and what you can do with it. So very briefly, Galois theory takes problems about polynomials and turns them into problems about groups. Roughly speaking, if you've got a polynomial, you can get a group out of it, which is called its Galois group. And what I'll do now is just give a few historic examples of this. So perhaps the most famous of them is the problem about can a polynomial be solved by radicals? Now there's one case of this, everybody knows. If you've got a polynomial a x squared plus b x plus c equals naught, you can solve it by radicals, meaning you can write down an expression for x in terms of a, b and c. x is minus b plus or minus the square root of b squared minus 4 a c all over 2a. And you notice this expression only involves the usual for arithmetic operations together with taking n-th roots. And you can ask, can you do this with higher degree polynomials? And the answer is you can do it for degree less than or equal to 4 and no in general for degree greater than or equal to 5. So this is the Arbol-Ruffini theorem. It's not entirely clear who proved it first. Ruffini had a sort of 500 page proof of it except no one's really quite sure whether it was a proof of it or not and sort of suspect that it wasn't. And a little bit later Arbol came along and gave a very clear six page proof of it. So it's a complicated problem for historians to figure out how the credit should be divided. Anyway, if you turn this into Galois theory, if you've got a degree n polynomial, you get a Galois group which is a subgroup of the symmetric group on n points s n. And it turns this problem about whether a polynomial can be solved by radicals into the problem about whether this subgroup is solvable. So solvable, if you remember from a group theory course, means it can be split into a billion groups in some sense. Here you notice a billion groups are of course named after Arbol. And the answer is for group theory, it's fairly easy to show that the group s 1, s2, s3 and s4 are solvable, but s5 is not. So if you find a polynomial whose Galois group is s5, then you can't solve it by radicals. Incidentally, the term solvable for solvable groups comes from this correspondence. It sort of refers to the fact that an equation is solvable by radicals if its Galois group is solvable. So we'll be covering that later on and explaining why an equation can't be solved by radicals if it is Galois group s5. And also we'll be showing how you can solve an equation by radicals for the degree at most 4 using the structure of the Galois group. If you don't know the Galois group, this is actually very tricky to do. I mean, it's really impressive that people manage to solve equations for degree 4 without knowing about group theory. So another classical example is trisecting the angle. Well, here we've got a polynomial. We might, well, we can trisect, try and trisect an angle of 60 degrees with ruler and compass. And here we've got a polynomial, and we might take the polynomial whose roots are cosine of 60 degrees over 3 and whatever its other roots are. And this turns out to correspond, you can find a Galois group of this polynomial and it turns out to correspond to a Galois group which is cyclic of order 3. And it turns out that being able to construct something with ruler and compass is very closely related to the Galois group being of order of power of 2. And as the Galois group of this is is not a power of 2, the shows will show you can't trisect an angle. On the other hand, there's another very famous example of this by Gauss, which was the construction of a hectare deca-gun, or a regular hectare deca-gun. This means a 17 sided regular polygon. And Gauss discovered you could construct this using ruler and compass at the age of about 19. So, in fact, he was so proud of this that he wanted a hectare deca-gun carved on his tombstone, but was talked out of it because the stone mason pointed out that a hectare deca-gun carved in stone would be absolutely indistinguishable from a circle carved in stone, so they wouldn't really be much point. Anyway, classifying a hectare deca-gun turns out to correspond to the following group. You take the cyclic group of order 17, and you look at its automorphism group, and this turns out to have order 16. And 16 is indeed a power of 2, which corresponds to the fact that you can classify construct this using a ruler and compass. So Galois theory was sort of more or less invented by Abel and Galois. And, you know, I mean everybody involved in this seems to have been incredibly precocious. So Gauss was doing this when he was 19. Abel was actually dead by the age of 25. So, you know, he he made himself world famous in inventing Abelian groups in Galois theory and so on in about the time most of us are just finishing a PhD or whatever. And Galois was even more precocious in that he was actually dead by age 20. So Abel died of tuberculosis. Galois managed to die in a duel of all things. The circumstances of the duel are a little bit murky, but it appears to have been a sort of argument with someone over a woman or something like that. And the really odd thing about it is nobody at the time seemed to have thought there was anything odd about Galois getting kills in a duel over a woman at age 20. Anyway, so what's the main idea of Galois theory? So the main idea is as follows. Suppose given a polynomial a n x the n plus plus a naught and let's say it as rational coefficients. There are extra complications you get if you look at fields other than the rational numbers that we'll talk about and what we do is we look at the fields generated by the roots. Alpha one up to alpha n. And what we can do is we can form the Galois group, which is all permutations of the roots of alpha one up to alpha n, preserving all algebraic relations between these roots. So the Galois group is a subgroup of the symmetric group Sn because it's a subgroup of all permutations, but can be smaller. So let's give an explicit example when it's smaller. So if I take the polynomial x to the five minus two, then we can write down the roots alpha one, alpha two, alpha three, alpha four and alpha five reasonably explicitly. So alpha one will be the usual fifth root of two, which will be a real number. But then you can multiply this by fifth roots of unity. So alpha two is going to be the fifth root of two times zeta, where zeta is a fifth root of one, and then we'll get the fifth root of two times zeta squared, and the fifth root of two times zeta cubed, and the fifth root of two times zeta four. And now you notice immediately that there are some obvious algebraic relations between them. For example, alpha one, alpha three is equal to alpha two squared. Alpha two over alpha one is equal to alpha four over alpha three and so on. So you can't do an arbitrary permutation of these and preserve all these relations. In fact, it turns out the subgroup of all permutations preserving all the relations you can think of as order 20 rather than 120, which is the order of Sn. It's actually the Frobenius group of order 20 if you happen to know about Frobenius groups. So this polynomial here has Gallo group of order 20. And I said earlier that we were going to show that polynomial is solvable by radicals if and only if its Gallo group is solvable. And this polynomial is obviously solvable by radicals. We can just take fifth roots and so on. And this corresponds to the fact that its Gallo group of order 20 is a solvable group, whereas the whole symmetry group S5 is not solvable. So the formulation of a Gallo group as being permutations of roots is the way people used to do things. Nowadays, we use a slightly more flexible formulation. So what we do is we take an extension of fields. We might take a field K contained in a field L and we will usually assume this is a finite extension, meaning that L is a finite dimensional vector space over K, but that's not really essential. And the Gallo group will be the symmetries of L fixing all elements of K. You might ask why don't we just define it to be this group of all symmetries of L and forget about the subfield K? Well, you could if you wanted, but it turns out to be a really convenient to have a sort of relative notion of Gallo group where you fix all the elements of K. So the idea is a major theme in Gallo theory is to sort of do things in steps. So you might have a field M containing a field L containing a field K and to prove things about M, you first of all go up from K to L and then you go up from L to M. So it's very useful to to consider two fields rather than just one field because this makes the proofs a lot easier. So a fairly typical example of this will be you take the real numbers contained in the complex numbers. So what we want is all symmetries of the complex numbers that fix all real numbers and it's not too difficult to see what those are. The Gallo group has two elements. One is the identity element and the other is complex conjugation. So you remember if you take a number X plus IY you can take its complex conjugate which is X minus IY and this preserves all field operations. So Z1, Z2, the complex conjugate of that is the complex conjugate of Z1 times the complex conjugate of Z2 and the same for addition and subtraction and nearly everything else. So complex conjugation is a symmetry of the complex numbers. So here the Gallo group just contains two elements. So it's the group of order two and this makes the complex numbers particularly easy to deal with if you know the real numbers because this this group is so small and easy. So we can state the main theorem of Gallo theory. This says suppose that K contained in L is a Galois extension. Well, what does this mean? Well, it means the size of the Gallo group, the number of elements, is equal to the degree of L over K. So this is just the dimension of L as a vector space over K and here we're going to assume that this is finite. Infinite Gallo extensions are a little bit more complicated. So we'll see that the Gallo group always has order at most equal to this and if it has order equal to that then we call this a Gallo extension. And the main theorem says that then subfields M with K contained in M contained in L correspond exactly to subgroups of the Gallo group. So if we can figure out the Gallo group, then we know absolutely every field that lies between K and L and as we'll see later in the course this often allows us to to answer questions about L such as where the with and where you can express every element of L and using radicals in terms of elements of K. So now what I'm going to do is I'm going to give a very very brief overview of some applications of Gallo theory. These are mostly going to be rather advanced applications that I'm not going to cover in the course. This is just to give you some idea of what people use Gallo theory for. So at the moment the really topical one is the Langlands program. So what does the Langlands program say? Well, it says very roughly that Gallo groups of fields L containing the rational numbers are something to do with modular forms. So a typical example of a modular form might be the famous discriminant function which is q times product over n greater than or equal to 1 of 1 minus q to the n. So the coefficients are going to be q minus 24q plus 252q squared. So now these coefficients are the famous values of the Riemannians tau function which is all sorts of weird properties. For example, tau m tau n is equal to tau mn if m and n co-prime which was conjectured by Riemannogen. So modular forms are very intriguing and interesting objects and you can spend an entire semester just explaining what they are and the Gallo at Langlands program makes the rather extraordinary claim that representations of Gallo groups of fields that means ways that Gallo groups can act on vector spaces are something to do with modular forms and explaining the meaning of something to do with is really difficult. You can spend a lifetime trying to understand just what this correspondence actually actually means. Never mind actually prove bits of it. Whenever you can prove bits of this correspondence, it generally leads to very very striking results. For example, Wiles proved Fermat's Last Theorem and the way he proved Fermat's Last Theorem was you take a potential solution of Fermat's equation and from this you get something called an elliptic curve. This bit was done by Frey. It's called the Frey curve and the elliptic curve has an action of the Gallo group of the rational. So from a solution to Fermat's Last Theorem you can get some sort of action of the Gallo group and then Wiles showed that from this you can get a certain modular form. So Wiles's main step was to go from an elliptic curve to a modular form. And he used the representation of the Gallo group to construct this modular form and most of his paper is essentially a study of of the action of this Gallo group and finally Ken Ribbit showed that this modular form you can construct from a solution of Fermat's Last Theorem is not possible. So this ends up proving Fermat's Last Theorem and somehow the really hard step of this involves studying Gallo theory. Another weird place Gallo theory turns up is it's a Gallo group turns out to be related or at least analogous to a fundamental group in algebraic topology. So if we've got an extension of fields this turns out to correspond to some sort of covering space. So a covering space might look like you've got some sort of topological space such as a circle and then there's another topological space lying above it which is locally an isomorphism. You see if you take some small point up here and take a small neighborhood of the point it maps isomorphically onto a small neighborhood of its image there. So you should think of a covering space as being something like a field extension. So in this case, I guess we should have the field mapping to bigger field L. Unfortunately, there's a sort of reversal of arrows that maps between fields go in the opposite direction to corresponding maps between covering spaces. Now the Gallo group of has something to do with the fundamental group of the base space. Let's take this base space to be S1. So it's sort of analogous to the fundamental group of S1. So you remember the fundamental group of a topological space mean you pick a point and you look at all ways you can map a circle into it starting at that point up to homotopy and you can compose these by following one loop and then composing it with the other loop. And this turns out to be analogous to composing elements of the Gallo group. And a little and later on we'll be talking about the algebraic closure. So for any field we will construct an algebraic closure and the algebraic closure turns out to correspond to a universal covering space. So the universal covering space is in some sense the biggest simply connected covering space you can think of. So for a circle the universal covering space kind of looks like an infinite helix. I haven't drawn this helix very well, but you can sort of think of a big helix like that. And the fundamental group would actually be the integers which sort of corresponds to automorphisms of this helix over the circle. In fact this relation between Gallo theory and fundamental groups is not just an analogy. Growth and Dick showed that they're both special cases of the same thing in some sense. So he invented a generalization of topology called etal topology and showed that in the etal topology the fundamental group of an extension of fields is more or less a sort of fundamental group of the base field or at least related to it. So fundamental groups in algebraic topology and Gallo groups in field theory turn out to be in some sense almost the same thing. So I'll just finish by mentioning an old classical problem in Gallo theory that still hasn't yet been solved. This is the so-called inverse problem. It's very easy to state. So given a finite group g is there an extension k of the rational numbers so that the Gallo group of k over q is g. Here I should say that k has to be a Gallo extension. I should have said that as well. And this is known to be true for many groups. For instance Shafarovich proved it for all solvable groups and later in the course we will prove it for abelian groups and it's been proved for quite a lot of simple groups but it's still a completely open problem in general. Okay so the next lecture will just be a sort of review of field extensions and algebraic numbers and things like that.