 Welcome back to our lecture series, Math 4230, abstract algebra two for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. So to let you remember where we are in this lecture series, so over the last several lectures, we've been developing the notions that we need from linear algebra in preparation to talk about field theory. So we've developed that for an arbitrary vector space, we've developed the idea that there's a well-defined notion of dimension that you have a basis. Basis always have the same size, the same cardinal, I should say, that is the dimension. That independent sets can always be expanded into bases, that spanning sets can always be pruned into bases, and so we have this well-developed theory of dimension and bases of a vector space. Importantly, if you have a field and you have some ring that extends that field, particularly if it's a field itself, then you can view that ring as a vector space over the subfield, and that's why we really were interested in developing this idea of dimension and vector spaces in the previous lectures here. So we've also seen that earlier in this lecture series, we've also talked about polynomials and factorization. We've discovered that factoring a polynomial over a UFD, call it D, is the same as factoring over its fraction field. So frack of D. If your polynomial, if its coefficients come from UFD, or if it comes over the fraction field, it really doesn't make much of a bit of a difference either. That's really a consequence of Gauss's lima there. So this allows us to move from unique factorization domains to fraction fields, and so we're very interested in the ring f of joint x, like so. Well, f of joint x is a ring that extends f, and so it's going to be an infinite dimensional, it's gonna be an infinite dimensional vector space, and so we're gonna start pushing all these things together. We're bringing together factorizations, particularly for polynomials. We have these vector spaces, and so now we are ripe, ready to start our formal discussion of field theory. But typically when one talks about field theory, the word field is not actually the one that's used. Typically we refer to something as Gaoua theory. Now some people are much more conservative when they talk about Gaoua theory, like it's only Gaoua theory when we start talking about the Gaoua group. Or some people are like, oh, it's only Gaoua theory when we talk about field extensions, which of course we are gonna talk about that right now. Some people are much more liberal, that is in terms of extensions, field extensions falls under Gaoua theory. Some people like Gaoua theory and field theory are synonyms of each other. That's the terminology we're gonna use, of course, in this lecture series. Now, another thing to make mention about polynomial factorizations, remember earlier we said that a polynomial has a linear factor if and only if it has a root. When you compare the factorizations of the UFD and the unique factorization domain, and the UFD with its field of fractions, that's the word I was looking for, the factorization is the same by Gauss's level. But if you get a larger field, it turns out the factorization could change, perhaps because you introduced a root. Like if you take the polynomial f of x equals x squared plus one, you can view this as a polynomial over q of joint x. It's irreducible in that setting because as it is a quadratic polynomial, all the way it could reduce is if it has a linear factor and the factors, linear factors only exist if you have a root. The roots are plus or minus i, rational numbers don't have that. But we could look at a larger field, like we could extend q into the complex number c, in which case then f of x can factor as x minus i and x plus i for which this factorization over the complex numbers has no equivalent over the rational numbers. So extending the field can change the factorization and that's very much why we're interested in the notion of an extension field. Let's define formally what we mean then of course by an extension field. It's a very obvious definition, but let's say it anyways, if you have a field f, we say that another field e is an extension field of f if f is contained inside of it. That's it, that's what an extension field is. We call f f for a field and e because it's an extension, it's also conveniently next to f in the alphabet, which is a follow in a convention we often use, adjacent letters when we describe similar things. And so an extension field e just means it's a larger field that contains the smaller field. Sometimes instead of an extension field, you could say that e is an extension of f, things like that, that's what an extension field is. Now, like I was mentioning earlier, since e is a field containing the subfield f, that means e is an f vector space. Since e is an f vector space, it has a basis as an f vector space and that basis has a cardinal and it's well-defined. That would then be the dimension of e as an f vector space. And this is gonna be denoted as e colon f, the so-called degree of this extension. Now, if you look at that notation, e colon f, that looks a lot like the index of a subgroup inside of a group. That, the similarity of notation is intentional right here. We actually want to look at that because when we start introducing Galois groups, we will see there's a relationship between the degree of an extension field and the index of corresponding groups. But I'm getting ahead of myself here. So if this dimension is a finite number, that is if e is a finite dimensional f vector space, we say that e is a finite extension of f. Otherwise, if e is an infinite dimensional f vector space, we say that e is an infinite extension of f. All right? Commonly when one talks about a field extension, so f extends, e extends f is what I meant to say. This is often denoted as e bracket, not bracket, e slash f in that situation. Now again, when you look at that, it's like e slash f. If I had written something like g slash n, you would think of like a quotient group g mod n or maybe a ring equivalent. That's not what we mean for fields. It looks like it could be like a quotient ring or something like that. But remember that for a quotient ring to be well-defined, you can only mod out by an ideal. And in a field, the only ideals are the whole ring, which if you mod it out, you get the zero ring. Or the zero ring, which if you mod out, you just get back what you started with. So you don't talk about quotient rings with fields. So even though the notation looks like a quotient ring, there's no ambiguity there whatsoever. And why do we make it look like a quotient ring then? Cause actually it's meant to resemble a quotient group. Like we said earlier, this kind of looks like the index of a group. This also looks like a quotient group. Now remember that the order of a quotient group is actually equal to the index. And so there's an analogy that's gonna happen that we won't see for a while. Some people actually write, they'll take e, e extend f, and they actually put cardinality around here as shorthand for the degree. That's not notation I personally use that often, but be aware that is sometimes used by other authors, other mathematicians, because they're taking, they're taking the analog between field extensions and group quotients to the extreme, which when we get deeper into Galois theory, it's gonna be quite natural why one might do that. Let's get a very simple example of a field extension right now. Let's take the rational numbers and we can extend it to the rational numbers, join the square root of two. And this is a field extension of degree two, I claim. And how do I know that? Well, that's because the set q adjoin square root of two, every element in that set can be written as a plus b times the square root of two where a and b are arbitrary rational numbers. And so clearly this is the span of the numbers one and the square root of two, like so. It's not too difficult to see that the set one comma the square root of two is an independent set as a rational, viewing them inside of rational vector spaces here. Because if it were literally dependent, that would actually force that the square root of two was a rational number, but even Euclid knew that the square root of two is not a rational number. That's a classic, very elementary proof there, just using divisibility of things. So this is an independent set, which then shows that the dimension of this field extension, the degree, is at least two because it's at least a two-dimensional vector space. But it then turns out since this is also a spanning set, it's gotta be a basis so we get the degree right here. Why is this a basis for this? Because after all, what is this set? Q adjoin the square root of two is defined to be the smallest field that contains the rational field and the square root of two. We'll actually give a more formal definition of it in just a second when we prove Kronecker's theorem. But just so you're aware, and just so you're aware, I mean, we'll prove Kronecker's theorem in the next video, not this one. So take a look for that one in just a second. But after we've proven Kronecker's theorem, it'll be a lot more obvious why this field is the same thing as this. Because this field on the left, Q adjoin the square root of two is the smallest field that contains Q and the square root of two. Now, clearly this does contain Q because you could just take A to be anything B to be zero. That gives you all the rational numbers. You also could take, you get the square root of two because you could set B to be one and A to be zero. And this is in fact a field, okay? How do I know it's a field? Well, clearly it's closed under addition, subtraction, and multiplication. The fact that it's, you have quotients is basically the following observation. If you take one over A plus B times the square root of two, if you multiply this by its conjugate, notice what's gonna happen here. In the denominator, you're gonna end up with A squared minus two AB, A squared minus two, excuse me, A squared minus two B squared. Like so, you end up with an A right here, minus B over A squared minus two B squared, and there is a slight argument that has to be made that A squared minus two B squared is never zero. I mean, after all, can a perfect square ever equal a perfect squared times two? This is basically now coming into Euclid's proof that the square root of two is irrational. So A squared minus two B squared where A and B are rational numbers can never, ever, ever be zero. So this thing is okay. This would belong to Q enjoying the square root of two. And so basically, because this is a vector space, let me show you the argument here. Because this is a vector space, there has to be a dimension that dimension can't be one because if the dimension was one, that would mean that Q enjoying the square root of two is equal to the rational numbers, but we know the square root of two is not rational. So it's gotta be larger than that. Like we said earlier, it's gotta be at least two, okay? But since things of this form do form a field because they have multiplicative inverses, this is a field that contains Q in the square root of two and so that's what this field has to be, the smallest such field, dimension two is the smallest it could be because you can't do dimension one, right? So there's some important arguments that are happening in that situation there. So this is in fact a field extension of degree two. I wanna give us another example before we end this video with our definition of field extensions. If you take Q as a subfield of R, we argued actually in the previous lecture here that as a vector space, R is a continuum dimensional vector space. That is the dimension of R as a rational vector space is equal to the continuum. That's this statement right here. So we can get lots of variety here. We can get these finite extensions, we can get infinite extensions, where to this case, the extension is in fact countable. And so the whole point of studying linear algebra in the previous lectures of this lecture series was so that we could utilize linear algebra to study field extensions, much like we did in this short video.