 Welcome back to our lecture series Math 42-30 abstract algebra 2 for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Missildine. In lecture 33, we are going to talk about the idea of a field automorphism. So up through now, I should say, up until now in this unit on fields, we've been talking about fields, field extensions, and all those type of things. Now we're really into the heart of Galois theory. It's not just about the field extensions, but how do these field extensions have symmetries? What are the symmetries of these field extensions? Well, how do you measure symmetry when you use automorphisms? Let me try to be a little bit more specific. Imagine we have some set X and there's some structure on the set. This could be some type of algebraic structure. It could be a ring, a group, a field, a quasi group, a semi group. There's a lot of options you could choose there if you want to do algebraic things. We could also require that there be some type of topology or geometric structure on the set. We could attach those things together. We could have something like a lead group, which is both geometric and algebraic simultaneously. We could have some type of topological field like the real numbers. There's some structure. We could put some combinatorial structure on the set. Maybe it's like a graph or order. Maybe it's a partially ordered set. There's a lot of possibilities on what we can do when we talk about a structured set. Whatever that word structure means in the situation, whether it's a group, a topology, a graph, whatever, there's some structure on our set. In that setting, what do we mean by an automorphism? An automorphism, as the word here means, shape and auto is self. Automorphism is measuring the shape of the object with itself. An automorphism on X is first and foremost a bijective function. It's going to be a function from X to X. Let's call this automorphism phi here. We say that phi preserves the structure of X. What does that mean? Well, in the case of a group, we would want to have the homomorphic property so that if we have the product of two things in the domain, this coincides with a product of two things in the range. That way we're preserving addition. We can preserve a multiplication. When it comes to algebras, we want to preserve the operations in play. If you're talking about a graph of some kind, like you have these points, something like the following, maybe we have the following graph. We'll call this one, two, three, sorry, one, two, three, four, something like that. We would want to preserve the adjacency of things. We might say that, oh, I'm going to take an automorphism that swaps one and two and swaps two and four. Because you'll notice that beforehand, we have that one and two are neighbors, but when you come over here via the automorphism, one goes to three, two goes to four, and yeah, three and four are still neighbors. The adjacency was preserved there. Let's look at another one. If we did something like say two and four, in this situation, two and four are not neighbors. They're not adjacent on the graph. Then when you transform them, well, two becomes four, four becomes two. They're still not neighbors. So adjacency is preserved. It's what we would call a graph homomorphism, automorphism. That's really what I meant to say there. And we could say similar things for a topology, a topology. We want to preserve the continuous structure of the sets. We want to preserve open sets of things. We look for what we call a homeomorphism. That one's a little bit more technical to get into for this lack, this algebraic lecture. So I'm not going to say much about that. But if we have some type of partially ordered set here, so we have some operation, so x is less than or equal to y, we would want that. This translates to that phi of x is less than or equal to phi of y. So a poset, automorphism is going to be a bijection that is increasing, an increasing map, an increasing bijection. We'd say that is an automorphism in that situation. And so whatever the category you're in, and I use this word category in a technical sense, for which there is a branch of mathematics referred to as category theory, for which a category, you take a collection of things, you take your collection of things, we'll call them the objects inside your category. Now technically speaking, this collection is not a set. It's a proper class, but I don't want to delve into the details there. So we have some collection of things, so you could have like the category of sets, in which case your objects are sets. We could take the category of group, for which then your objects would be groups. We could take the category of partially ordered sets, in which case your objects are posets, exactly like we have there. Then we're going to take a collection of things that we think of as functions, for which a and b are two objects that belong to this category. And we take all these so-called morphisms, as they're often called. Some people call them homomorphisms, but I think in category theory morphism is the more common word there. So we think of these as functions between the objects, but as the objects themselves don't have to be structured sets, these morphisms are not necessarily functions themselves. But in the context of where we're right now focusing on, we're talking about structured sets. So we're looking at categories of sets, which have other structure to it. Amorphism then, in that case, is going to be a function between a and b, which preserves. It's a function which preserves these properties, for which, like we said, for the partially ordered sets, we want increasing functions. Those are morphisms. With groups, we want homomorphisms, things that preserve the multiplication, etc., etc. For a topological space, we want continuous functions. These are going to be our morphisms, and it depends on the category. And so when it comes to this category theory, there are, of course, a few rules in play here that they often denote these sets as hom of a, b. So that's this thing right here, hom of a, b. This is the set of homomorphisms from a to b, like so. We do have some requirements that if you take the hom of an object with itself, it's never empty. So these, these homes are back sets. Even, even though the collection of objects is not necessarily a set, these homes are always a set for any two objects. This is always a non-empty set. And that's because it always contains the identity, the so-called identity morphism. Now, if you think of these as functions, this is the usual identity map in the usual sense. We also require compatibility and associativity law of some kind. So if f belongs to hom of a and b, and if g belongs to hom of, let's say, b, c, where a, b, and c are different objects here, we are then going to get that when we compose these things together, there is, there is some object g composed with f, which belongs to hom of, the hom of where b, a to c here. And this, this, this homomorphism is well-defined. That is, we have a composition rule. And then you can say that this composition rule is associative. This element, this guaranteed element, the identity acts like an identity element, right? If you take the identity composed with f, this is the same thing as the f morphism, what have you. So you don't have a, you don't have a group, because there's no guarantee of inverses with this axiomatic system for category theory. This is actually where the word monoid comes from, by the way. A monoid, remember, is a, it's, it's, you have an associative binary operation with an identity. It turns out, the reason is called monoid, because after all, doesn't mono mean one. A monoid is actually a one object category. That is, if you have only one object in your category, then there's only one homset from a to a. It's not empty because it has an identity. And because composition of these morphisms is associative, you can then think of the elements of the homset. They form a monoid in the algebraic sense. So a monoid is, it gets its name because it's a one object category. A little bit of a tangential discussion there, but I want you to be aware that this is what we mean in the structured set. We're thinking of it a categorical approach. We have these categories of sets with different structures. And so the homomorphisms, the morphisms depend on the category. Are they homomorphic? Are they continuous? Are they increasing? It depends on the category. And so with that, we can talk about things like, what's a monomorphism? This would be an injective morphism. What about an epimorphism? Well, that's going to be a onto morphism. What's an isomorphism? It's going to be a bijective morphism. Again, when these morphisms are functions. In the more general sense, they don't have to be functions. And I can make precise what these definitions are, like what does it mean to be injective or surjective, bijective in that sense. But we won't delve into that too much. It also makes sense to talk about an endomorphism. And endomorphism are those things that belong to the homsets a to a. And thus putting those all together, we get this idea of an automorphism. An automorphism is going to be a bijective morphism of the category. It goes from the set back to itself. It is bijective. So it forms an isomorphism from the object to itself. And based upon the categorical structure, it has to preserve whatever that is. And so in any category, we get these idea of an automorphism group for which we can take the set of all automorphisms on an object X. So with the language we were using before, it's not quite harm of a comma a, because it's really a subset of that. That looks hideous. Try that again. It's going to be a subset of that because we're looking for those invertible morphisms inside of that. This forms a group in the natural way where composition is your operation, because the identity will of course be bijective. By assumption, the composition will be associative. And then by assumption, these are the bijective elements that always forms a group. So this is one of the true strengths of group theory that no matter where you are, no matter which category you have, you can have the most bizarre category in the world, it's natural to introduce groups into that setting, because groups measure the symmetry of those objects, however bizarre they are. So group theory can permeate anywhere mathematics because anywhere mathematics, no matter what your category, symmetry could be important. And the symmetry group, aka the automorphism group, is in fact a group. All right. Now, in addition to that, let's suppose that we have our structured set X. And suppose we have some type of subset, right? It's going to be a structured subset. So we think of like a subgroup, it's a group inside of a group. Or we could think of like a subspace, a topology inside of another topology. We could do that with partially ordered sets as well, like the rational numbers, which is an ordered set, lives inside the real numbers, which is an ordered set. So we have these substructures as well. And we can define a substructure as there exists some monomorphism from one into the other, injective morphism from the subset to the other set. We can then make that into a structured subset. Well, sometimes it's of interest with our automorphisms to consider the automorphisms of X, which fix Y. That is to say, we have some fee that lives inside of the automorphism group of X. We then have the restriction that is we have the assumption that when you restrict the domain to the subset, this is just the same thing as the identity on Y. So we can talk about those automorphisms, which fix a certain structured subset. And we're going to note that as the automorphisms of X, Y. Now, there's two ways people often write it. Some people write it use a slash here. So X slash Y, some people use a comma, both are used quite regularly in the literature. I feel like this one's probably a little more natural because it kind of feels like a quotient structure of some kind. It's kind of mimicking that. But be aware that it's not, you can't see things like this sometimes, all right, where it's like X comma Y. So this set right here, it denotes all the automorphisms of X that fix Y. Now, if Y is equal to X, this would only be the identity map. So this thing is always not empty. But depending on how big Y is compared to X, there could be a lot or a few automorphisms in there. Now, suppose that we have two elements, two automorphisms, phi and pi that belong to the automorphisms of X, but they fix Y, right? Be aware that when you compose two automorphisms together, that gives you an automorphism. But if both of these automorphisms fix Y, take an arbitrary element little Y inside of Y, notice here that if you take phi of pi of Y, pi will fix Y since it's inside of this set, and then phi will fix Y because it belongs there too. So this shows you that the product of two automorphisms that fix Y will also fix Y. So this is in fact a subgroup of the automorphism group. So if you have a substructure inside of your structured set, the automorphism group naturally gets these subgroups as well. It will contain the identity, of course, because the identity fixes everything, let alone subsets. And you can also argue by similar reasoning that the inverse of an automorphism that fixes Y will also fix Y. Because if the elements of Y are fixed, if you reverse the directions, they'll still be fixed. And so this object that we're describing right now, these automorphisms is going to lead to the idea of a so-called Galois group. That is, a Galois group is going to be a automorphism of a field extension E over F, right? So we think of E as an extension of F. The Galois group is going to be those automorphisms of the larger field E that fixes the subfield F. And so while we can do these, we can do these automorphism groups in a general context, what our goal is in this lecture series right now is to go into field extension to the field theory and look at these automorphisms in that context.