 In this video, I want to establish and state the so-called fundamental theorem of Galois theory, which as the name suggests, it's a big deal, right? It's going to be the culmination of basically two semesters worth of abstract algebra, combining notions from field theory, ring theory, group theory, all together under one big umbrella. It's a fantastic result. And honestly, Galois theory is one of the reasons why we have modern algebra in the current sense, the works of Galois were incredible, and it's a privilege for us to go through these things together. Now, before I show you the statement of the theorem, it's important that we understand what we mean by the Galois correspondence for which the fundamental theorem is all about. And the idea is that we have these two categories of objects. We have over here our fields, and we have over here our groups, okay? And so our problem is usually motivated by the following situation. We have some field extension E over some base field F, like so. We can turn this into a group by taking the Galois group Gal of E over F. Okay, for the sake of simplicity, I'm going to call this Galois group curly G. And then we have our field extension E over F. So we have this, we can do this, we can turn this field extension to a group, and this group measures something about the field extension, like when this is a Galois extension, we find out that the order of this group is equal to the degree of the extension. You only get an inequality in the general case, but that's very important as we've been going forward with this. So what we want us to do is consider, we look at the subfields, the subfields of E, but the subfields that contain the base field. So we're looking for fields K that are going to sit in between E and F, which these fields do not have to be proper. This would include E itself, this would include F itself, but we also contain everything in between them. So we have all these fields. These fields K can likewise be turned into groups, right, where you're going to get this the Galois group of, in this case, E over K. Like so that is we could take all the automorphisms of E that fix the field K. Now because K contains F, anything that fixes all of K will also fix all of F. And so this is naturally going to be a subgroup of the Galois group. So here's this correspondence between the subfields of E with the subgroups of the Galois group. Okay, we can turn a subfield of E that contains F into a subgroup of the Galois group of E over F. And so we can think of this operation here, this operator Gal of E over dot, okay, where that dot we can insert the subfield we want to, there's this operation that turns the subfields into the subgroups. So this process in fact, that we can go the other way around that if we start with the subgroups, that is we take some subgroup H that is contained inside the Galois group, we can then turn this into a subfield. We can turn that into a subfield by taking E sub H, where this is the fixed field, the fixed subfield of E with respect to H here this is the set of all things inside of E so this still sits inside of E. This is the set of all things inside of E that are fixed by the on morphism group H. We've proven that previously this is a field contained inside of E, and clearly it'll contain F as well. Because everything in G fixes F. So if I take some subgroup it's still going to fix F. But by taking smaller on morphism groups you potentially are getting larger fields like so. And so this is this is a very important observation. Well, before we go on though this map where you reverse the direction you could take a subgroup and turn it into a field, we can think of this as the operation E dot down here. We have this fixed field operation that turns on morphism groups into fields, and then above we have this Galois group operation for which we can turn fields into on morphism groups. And it turns out the fundamental theorem of Galois in a very simple sense says that these two operations are inverses of each other. You have this one to one correspondence between the subfields of E that contain F and the subgroups of the Galois group. And this is then referred to as the Galois correspondence. We have this Galois correspondence. Now this Galois correspondence, it is a one to one map, but it's not just a map between these elements, there's actually lattices, like over here we have this lattice of subfields that as we could draw it. We could draw a haza diagram right there's this partially ordered set. And likewise over here we have this lattice of subgroups. And so this Galois correspondence is not just a bijection between subfields and subgroups. It's an isomorphism between the lattices, but it's a little bit weird in that it's an order reversing isomorphism order reversing. So things the containment gets flipped upside down with this correspondence so big coincides with small and small coincides with with big here. And so actually when it comes to this type of phenomenon when you study lattices, if you have an order reversing isomorphism, oftentimes it's referred to as a Galois correspondence because we recognize it's just the generalization of what the fundamental theorem of Galois tells us. Now let's look at the statement of the fundamental theorem of Galois theory. So suppose that we have a Galois extension E over F. Now typically for a class like math 4230. If we were going to prove this we'd assume that this is a finite Galois extension. Infinite Galois extensions get a lot more complicated. And so the proof is definitely in that case would be on the scope of this course. So we can assume that E over F is a finite Galois extension. But most of the statement is still true even in the infinite case. Be aware once you once you attach the appropriate Zernsky topology to take care of things. So we have a Galois extension E over F. Consider the Galois group of E over F which will abbreviate this as a G. So again for the state for the purposes of this statement of the fundamental theorem of Galois, we're going to assume that the Galois group is a finite group. And for the infinite Galois extensions analogues do exist that we will not discuss further. And so there's actually multiple parts to this and the fundamental theorem of Galois is going to establish this Galois correspondence that I hinted towards on the previous screen here. So imagine that we have two fields that sit in between E and F. So K and L are both subfields of the field E and they contain the base field F. And suppose that K actually is a subfield of E as well. So we mentioned earlier how oh yeah you can take these subfields and you can turn them into Galois groups by taking Gal of E over something. Okay. So the first statement here says that if K is a subfield of E, and particularly if you have this chain right here, F is contained inside of K, contained inside of L, contained inside of E as fields, then these will all turn into Galois groups by the Galois operator here. But the order gets reversed, right? The very smallest field F then becomes the largest Galois group. Galois of E over F is larger than Galois of E over K, which is larger than Galois of E over F and Galois of E over E, which of course this very last one is just the trivial group right there, but still worth mentioning. So the Galois operator, that is the computation of the Galois group over a chain of fields, reverses the order of containment. The biggest field coincides with the smallest subgroup and the smallest field coincides with the largest Galois group. And when you have intermediate fields, if K is inside of L, the Galois group of K contains the Galois group associated to L in that situation. So this is very important here that the Galois correspondence is order reversing. The Galois group operator switches the directions of containment. And then part B here, it goes in the other direction. Suppose that we have our Galois group G, clearly at the very bottom of the lattice of subgroups is going to be the trivial subgroup, the trivial subgroup, just the identity, right? And suppose we have two subgroups that sit in between. So we have this chain of automorphism groups. The trivial group is contained inside of H, which is contained inside of K, which is contained inside of G. And so like we mentioned, that the Galois correspondence is a bijection, it's reversible, okay? So we have this fixed field operator that can send groups into fields, into the associated fixed fields. This correspondence, this operator is also order reversing. That is the trivial subgroup sits inside of everything. And the fixed field associated to the identity contains everything because after all, this is just E, right? What is the subfield of E that's fixed by the identity? Everything, literally everything is fixed by the identity map. And so that E1 is just E that contains everything, okay? On the other extreme, if you take G, which is the largest Galois group, what is fixed by G? If your Galois group, by definition, it fixes F. And you're going to have different automorphisms with these different fields. The only thing that can be uniformly fixed by every automorphism is in fact F itself. This fixed field EG is just going to be F itself, alright? F is the only field that's fixed by everything that fixes F. And the intermediate fields, excuse me, the intermediate subgroups, H is contained inside of K. This will translate so that the fixed fields, we have that EH contains EK, alright? So this fixed field operator, in fact, is also order reversing. It sends a chain of groups to a chain of fields, but the order got reversed now. And I've referred to these operators as inverses to each other. That's what the next two statements of the fundamental theorem of Galois there tell us, that if you have some subfield K inside of E, and of course it contains F, and you look at the associated Galois group, the Galois group of E over K. So this is the set of automorphisms of E that fixed K, let's call that H. And then if you look at the fixed field associated to H, that gives you back K, right? So if we expand upon this a little bit, you have E sub the Galois group of E over K, that's just going to equal K. So if you apply the Galois operator, then the fixed field operator, you get back the original field. So in that regard, the composition gives you the identity map on the subfields, okay? Which then if you look to part D here, we get the other containment. Suppose that we have a subgroup of the Galois group H is a subgroup of G, and then you take K to be the fixed field associated to the subgroup H. Then it turns out that the Galois group of E over K was actually H to begin with, which again, let me expand upon this a little bit. We're saying that the Galois group of E over K, but what's K? K is the fixed field associated to H, this is equal to H. All right? So this tells you that if you start with a subgroup, if you take the fixed field associated to that subgroup, and then you take the group of automorphisms that fixes that subfield, you get back the original subgroup itself. So again, the composition of the fixed field operator with the Galois group operator gives you back the original here. And so what we see here when we put these things together is that with regard to the set of subfields of E that contain F and the subgroups of the Galois group, these two operations of Galois group versus fixed field are inverses of each other. That if you do one to the other, you get back the original object, whether it's a subfield or whether it's a subgroup, they're inverses of each other. So we in fact do have, there's this one-to-one correspondence between the subfields of E that contain F and the subgroups of the Galois group, but this correspondence is order reversing. It switches the directions of the inequalities here. And so these four principles together establish this Galois correspondence, this bijection between the lattices that's order reversing. Now it turns out that this Galois correspondence is much more powerful than just we send subfields to subgroups in an order reversing way. This last principle here, think of the following. Suppose we have a subfield K inside of E that is normal over F, okay? So K over F is itself a normal extension. This happens if and only if the Galois group of E over K is normal inside of G. This is actually established in the fact that we called normal extensions normal. It was foreshadowing to this Galois correspondence so that the normal extensions inside of the lattices subfields coincide with the normal subgroups of G. And because it's a normal subgroup, that means we can quotient it out. It turns out that if you take the Galois group of E over F, so that's the whole thing, that was G. And you can take the normal subgroup Galois of E over K, because again, if E over F is a normal extension that makes this into a normal subgroup of the Galois group, you can quotient it out. This as a quotient group will be isomorphic to Galois of K over F. And so this is a fantastic useful result that the normal extensions coincide with the normal subgroups. So this corresponds as much of a stronger. It's more strong than just subfields correspond to subgroups. We get that normal extensions coincide with normal groups. Again, this all happens inside of a Galois extension. It's fantastic. It really, really is. And what's the proof of this statement here? You'll notice that nothing's listed down below. Well, I'm omitting the proof for a couple reasons. One, a lot of things we've already proven we have. Like statement C was something that we've already proven. I think in the previous video we proved this one, that if you take the composition in that direction, they cancel each other out. Okay. We also proved some of these containment symbols. If we didn't prove them exactly, we've alluded to it. So we've proven a lot of these things already. Now, officially speaking properties, parts A and B, I'm actually leaving formally as homework exercises to my students. So I encourage the viewer to try to prove them on your own. Like I said, C, we've already done. D does take some more work. We haven't established this one yet. And then E is definitely a doozy compared to the rest of them. That one takes a lot of work. And so I'm making just a judgment call here that while the proof of the fundamental theorem of Galois is not beyond the scope of an undergraduate abstract algebra class, particularly in the second semester, it does come down to timing. And that shows that you're watching are connected to an actual class taught at Southern Utah University. And so this class because of our semester schedule contains 39 lectures and so in order to cover everything. Some sacrifices have to be made. In fact, I really can't cover everything. There are some choices that have to be made. So here are one of the things I want to my students to be exposed to. And I definitely want my students to see the fundamental theorem of Galois. And while the proof of its entirety is quite capable, it is time consuming. And I feel, especially since we're at lecture 34 right now, like I mentioned a moment ago, we are semester ends at lecture 39. Some of the sacrifices I made was to skip over the proof of the fundamental theorem of Galois. And there's actually a lot of results in Galois theory that we are not covering right now. A notable other result that we aren't going to even we're not going to state improvement this lecture series is the so called primitive element theorem, primitive element theorem, for which this theorem tells us that if E is a finite separable extension over F. So E over F is finite and separable. So remember to be a Galois extension, you are a normal separable extension. So if we have a finite separable extension here then there exists an element alpha inside of E, such that we get that E is actually equal to F over alpha. So with these finite separable extensions, it turns out there's a single element that actually generates the whole thing right there. This primitive element theorem is an extremely, extremely important, very useful result in field theory. But for the sake of timing, we're skipping over it. Okay. Likewise, like I said with the fundamental theorem of Galois, I want us to focus on the statement so that we can then get to computations of Galois groups. But be aware, we're not going to develop all of the theory here. Later on, in lecture 36, we are going to prove the insolvability of the Quintic equation. That's perhaps the most important application of Galois theory because that's what motivated in the first place. So I'm definitely going to cover that one. But another notable result that would be very awesome not beyond the scope of this course would be to prove the fundamental theorem of algebra. The fundamental theorem of algebra remember states that every complex polynomial has a complex root. In other words, the field of complex numbers, C, is algebraically closed. That's what that theorem states in the language of field theory. We can prove it. It's not beyond the scope of a lecture series like Math 4230. It's a pretty fun application. It's just it does take time, time that we can't dedicate to everything. And if this was a graduate course, by all means, we would do all of these results. And some other ones I'm not even alluding to, like the infinite Galois extensions. We would say something about that because in the graduate Galois theory class, that's exactly the type of stuff we need to cover. Students need to see these things. But let's be honest, I don't think there's ever been a student that's ever existed that perfectly understood Galois their first attempt through. It's a very difficult concept compared to many of the other things we've done in abstract algebra. This is a challenging discipline for many students to begin with, let alone the hard parts of abstract algebra, aka Galois theory here. So we're at a very important point that many of the students who would be participating in class like Math 4230 will go on into graduate school and have more exposure to abstract algebra, particularly they'll have a second, a second coverage of Galois theory, where a lot of these harder results will be stated and proved in entirety, including of course the fundamental theorem of Galois. And so what, why are we giving up these results then if they are within our purview. Well, the idea is that I want us also to be exposed to many things that often aren't covered in a theoretical graduate level abstract algebra course. Many of the applications we've seen in this series aren't covered in those graduate courses typically not. And so this undergraduate course is the time for us to see those things. Things like cryptography coding theory, just to name some of those some of those things that they took some time like for this lecture series, the very last three lectures are going to be on the topics of lattices and Boolean algebra, which have huge applications, but are often skipped over in graduate school, modern algebra classes because the focus is on groups, rings, Galois theory and that you know when you want to cover everything you got to give other places. So for this, for this lecture series I've intentionally given time for these topics that are very important. That aren't necessarily covered in graduate school and the reason I do that is because well those students who go on the graduate school, I want them to have some exposure to these topics in algebra that they might not see later on, but might see in the future not in graduate school in the future, they should have some exposure. And then also I recognize that many of my students and many students of algebra, who might be in a class like this one math 42 30 aren't going to go to graduate school or maybe later on but they're not necessarily planning to go to graduate school right now, they might be getting into education they might be their careers might be into applied mathematics or computer science or data science or something like that, for which case a lot of these applications like cryptography coding theory Boolean algebras have huge huge applications in applied math, particularly in computer and information technology. There's a lot of applications there and so I feel like the, the, the more equitable approach to a lot of these Galois theoretic topics is to defer some of these more advanced results to optional videos, and then the required lecture courses being focusing on the things that we have right here. So all of these topics, I just mentioned the proof of the fundamental theorem of Galois primitive element theorem fundamental theorem algebra. These are videos that, if not presently will be available as optional videos that you can check out on this channel, but be aware they're not required for the students in my class, because of this timing constraint that that I've just discussed. So these results are very, very important. And feel free to look for the resources of you if you do need to see those proofs. But for the purpose of my students I really want them to focus on the understanding the statement of the fundamental theorem Galois, and then we're going to use our time to do some computations of Galois groups, which we'll see in the next several videos.