 Welcome back to our lecture series Math 4230, Abstract Algebra 2 for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misseldine. Lecture 30 is going to be continuation of what we did at the end of lecture 29 naturally, for which we had started talking about the idea of a finite field. So you'll recall that at the end of lecture 29, we proved that all finite fields will have two very important properties, that the order of a finite field is always a power of a prime. And that's often abbreviated as Q, right? So when you see Q being used as a subscript, that means the power of a prime. So all fields have an order, all finite fields have an order of power of a prime. And in fact, if you have two finite fields, let's say that f and e are finite fields who have the same order, they both equal Q here. That's a hideous Q, try that again. Then in fact, the finite fields are isomorphic to each other. So finite fields are unique up to order. This is kind of the same thing for cyclic groups. In many ways, finite fields are the field equivalent of cyclic groups, finite cyclic groups, for which we'll see some more of that parallel later on in this video, in fact, well, not this video, actually the next video in this lecture series, in this lecture in particular. But in this video, we are going to define the notion of the Gawa field. What does that mean? So let Q be a power of a prime. So Q equals P to the n where P is a prime. Then we know by the previous result that there is one field up to isomorphism of order Q. So we call that the field of order Q. It'll be denoted often as FQ. Typically the blackboard font is used if you put in latex, but by hand sometimes you'll write it just F sub Q. Some people will often denote this as GF of Q. Now that does not mean gluten free in this context. We call it the Gawa field named after a Gawa, of course. And so commonly when people refer to finite fields, you might hear them called Gawa fields as well. So finite field and Gawa field are used interchangeably in this situation because as there's a unique finite field of order Q, we call that the Gawa field of order Q. Now this is not to be confused with the Gawa group, which we'll define later on. The Gawa group does measure an important attributes of field extensions. It is related. So we can talk about Gawa groups of Gawa fields. And this is also not to be confused with the term of a Gawa extension. Because of that potential confusion, I think I'll prefer just to call them finite fields and the field of order Q as opposed to call them Gawa fields because I don't want to confuse with the Gawa theory which we'll do a few more lectures later, but be aware that is used in practice in the literature Gawa fields are referring to the finite field of order Q. All right, so let me prove a very important theorem when it comes to Gawa fields. In particular, how are they nested inside of each other? So given any field, so if F is a field of order P to the N, right? Then that tells us that ZP is in fact going to be contained inside of F because this is the prime field. Now ZP is by definition the field FP, right? And this is the field FQ. So every finite field is gonna contain its prime field. But what about other ones? We've talked about previously in this lecture series, the field of order four. There's also a field of order eight, order 16, order 32, order 64. Do these fields contain each other? Is it like the field of order eight contains the field of order four and the field of order 16 contains the field of order eight? That would lead to a lot simpler containment but it's not like that. The webbing is a little bit more complicated and it has to do with the divisibility of the exponent. And that's what the theorem we have on the screen tells us. So let P be a prime, then the field of order P to the M is contained inside of the field of order P to the N if and only if the exponents M divide N. And so this is an if and only if statement will go in both directions. So assume the first direction. Suppose that one Galois field is contained inside the other. So we have P to the M contained inside of the field of order P to the N. Now this field containment, this field extension implies a vector space argument as well. That is this is a, we can view this as a subspace of this vector space. Both of them are vector spaces over the base field FP. So when we look at F sub P to the N, this is an N dimensional FP vector space. And when we look at FPM, this is going to be an M dimensional FP vector space. But because this is a field extension, the degree of the map FPN over FP factors as FPN over FPM by assumption that's a subfield and FPM over FP for which, like I said, this first one is equal to N. This one is equal to M. And so then we see that N can be factored using M and that's exactly what divisibility means. So M does have to divide N to be a subfield. Okay, why does it go the other direction? Well, suppose that M divides N, that is there's some integer K such that KM is equal to N, like so take an arbitrary element of the field F sub P to the M. And I want you to consider the polynomial X to the P to the N minus X. Notice here that by previous work, the field of order P to the N is the splitting field of this polynomial. So, and we actually showed a stronger argument. You belong to the field of order P to the N if and only if you're a root of this polynomial. Okay, and so that's what we're gonna do. We're gonna show that alpha is a root of this polynomial. Therefore, it is contained inside of the splitting field of that polynomial. So therefore, since alpha was chosen arbitrarily, the entire set F sub P to the M is contained inside the field of order P to the N. That's the strategy here. So let's evaluate the polynomial F at this element alpha. Now, we do have the property that since alpha belongs to F sub P to the M, we have that alpha to the P to the M power is equal to alpha. Okay, again, this was something we proved in the previous video here. So if we take alpha to the P to the N minus alpha, we can factor N because N is equal to K times M, like so. And then using some exponent laws, this is the same thing as alpha to the P to the M to the Kth power for which what does that mean here? So we have to be careful with our exponents here. We're taking P to the M to the Kth power. We're not taking this to the Kth power. That's a different thing there. So in particular, P to the M shows up K times in the exponent. And so by exponential laws, if you have a factored exponent, you can iterate the exponent. So this is the same thing as alpha to the P to the M to the P to the M to the P to the M to the P to the M all the way up to the P to the M, like so. But by induction, since alpha to the P to the M is equal to alpha, this then becomes just alpha. Race to the P to the M just becomes alpha again. Then you have the next one, you just get alpha again and you iterate this until you eventually just end up with a single alpha. Alpha minus alpha is equal to zero. And so like we mentioned above, since alpha is a root of F of X and since the field of order P to the M is the splitting field for F of X, that means alpha has to belong to that. Since alpha was arbitrary, that means this entire set, F sub P to the M is a subset of P to the N and that then gives us the containment. So we get divisibility, divisibility implies containment. The field of order P to the M is a subfield of P to the N if and only if, right? If and only if M divides N. So let's look at a specific example. So let's look at the field of order P to the 24 where P is an arbitrary prime because the argument actually didn't depend on the prime, it depended on the exponent here. So if you take the field of order P to the 24, what are the divisors to 24? You have one, two, three, four, six, eight and 12 and then 24 of course. So the subfields of P to the 20, the field of order P to the 24 is gonna be these fields, one, two, three, four, five, six, seven, eight. So it has eight subfields because there are eight divisors of 24. All right? How are these fields contained inside of each other? So if you take, for example, 12, what divides 12? Four divides 12, six divides 12, two divides 12, three divides 12 and one divides 12. All the divisors are 24 divide 12 except for eight and 12, 24 itself. So those are the subfields of 12. Who divides eight? Well, four divides eight, two divides eight and one divides eight. You can't have any divisors of three. So those are the subfields of the field of order P to the eight. Who are the subfields of P to the six? Well, you get two, three, one and, of course, six itself. Who are the divisors of P to the fourth? Well, the divisors of four are four, two and one. And then since two and three are prime numbers, the divisors of P squared and P cubed are themselves and then ZP, FP here. And so FP, of course, is only divisor as well. So when you look at the lattice of subfields for F sub P to the 24, you see just the divisibility lattice of 24. So this lattice is isomorphic to the divisor lattice of 24. And so when it comes to organizing the subfields of finite fields, we get a very simple observation. This, the lattice of subfields of a finite field is just the same as the divisor lattice of the exponent of P in that situation. So this is our first look, of course, at Galois theory right here. That we have these Galois fields we talked about. The Galois theory is very interested in classifying these subfields of a field. Basically we look for lattices much like this. The Galois group then helps us organize these things. Now it turns out that Galois theory for finite fields is trivial because of the previous theorem. And so when one talks about Galois theory, we often ignore finite fields because it's such a special case that it's somewhat insulated from the rest of it. But in other words, we have solved this problem of the Galois theory, the Galois problem without even defining what the Galois problem is yet because finite fields are so well behaved compared to perhaps field in general. As we study Galois theory later on in this lecture series we'll be primarily interested in looking for the subfields of fields that extend the rational numbers. And that's something we'll talk about, of course, in a future lecture.