Algebraic Extensions (Pt. III)
Point of Post: This is a continuation of this post.
The above illustrates an important fact:
Theorem: Let be a finite extension. Then, every element of is algebraic over .
Proof: By the same argument as above, if then are distinct elements of an -dimensional space and so must be -independence, and so there exists such that or satisfies .
What we’d like to prove is that there is a partial converse to the above, which will roughly say that a finite extension of a given field is precisely a ‘small’ extension by algebraic elements, with whatever ‘small’ will mean. Why partial? Clearly not all algebraic extensions are finite since, for example, is algebraic and certainly is not finite (in fact, it’s )–but of course the difficulty here is that whatever ‘small’ will mean, we must have that is ‘large’.
To properly phrase this converse we need to discuss the notion of finitely generated algebras. Let be a commutative ring and an -algebra. We say that is finitely generated if there exists such that . If for some we call simple and call a primitive element or just primitive.
Of course, if is an extension we call it a finitely generated extension if is a finitely generated -algebra and we similarly define a simple extension.
This notion of ‘finitely generated’ is precisely the notion of “small” that we are looking for. To begin to show why then (substituting ‘finitely generated’ in for ‘small’ in our preceding paragraphs) finite extensions are precisely finitely generated algebraic extensions let’s show that the one way is true–let’s show that all finite extensions are algebraic and finitely generated. The first of these has already been done and since evidently for any in some extension of we can clearly conclude that implies that is finitely generated by taking the generating set to be the basis for as an -vector space. Thus, we may definitively conclude that
Theorem: Let be a finite extension then is algebraic and finitely generated.
So, how exactly should we go about concluding the converse? The idea is simple enough. Basically we are starting with a field sitting inside some extension , and we know that if we take some algebraic then has finite dimension over . Ok, so what happens when we add in another ? Well, there is nothing a priori that we can say about the dimension of since we only know things about appending roots to itself. The key in thinking (and this is an important idea, so keep it in your back pocket) is that instead of thinking about as appending both and to let’s think about appending to and now think about everything in sight as an extension of . Why is this the right way to go? There are two reasons for why not only is this a good idea, but a natural one. First, all of our theorems up until this point have concerned us appending a single element to a given field. Second, thanks to our product formula we know that if we can cut the extension into a tower where each piece is finite then the top field is actually finite over the bottom one, and since analyzing a tower piece-by-piece should be easier, it should always be our first instinct.
Ok, so because of the above we have decided that the first natural thing to do to decide whether or not is finite is to examine the tower
and decide whether field is finite as an extension over the field immediately below it. But this is easy. Why? Well, we know that and so if we know that is algebraic in we’re golden. That said, it’s trivially algebraic since we know that and . In fact, we know that is but since annihilates we have that so that, in fact, . Thus we have proven the following:
Theorem: Let be an extension and algebraic in . Then,
Thus, we may definitively conclude the following:
Theorem: Let be an extension. Then, is finite if and only if it is algebraic and finitely generated.
While this theorem and the theorems leading up to it may have seemed to be mostly a digression from algebraic extensions/elements to a tangent about finite extensions it actually has a very important use. Namely, suppose that we have some extension . Up until this point we have been looking at individual members of the set of all algebraic elements in instead of the conglomerate of all such elements. In other words, if we let be , called the algebraic closure of in , then we have been looking at individual elements of instead of itself. That said, it seems fairly natural to want to know things about itself. Since (obviously) probably the most natural of these natural questions is to whether is actually a subextension of , or said differently, is a field?
As is (unfortunately) often the case, this most natural of natural questions is not very obvious. Think about what might be the obvious route to proving this. We’d take some and some –good start–then we’d say that there exists such that –right, good, good– and then we just…. Just what? There is no obvious way to take the polynomials and and use them to make some new polynomial such that . For example, the minimal polynomials of , and over are , , and respectively. That said, one can verify (via, say, wolframalpha) that the minimal polynomial of is and the minimal polynomial of is
So, in general, it seems pretty hopeless to easily be able to prove that the sum of two algebraic elements is algebraic via the direct method of finding a polynomial which annihilates. But, using our above machinery we can indeed prove that is a field very easily. Indeed, we want to prove that , right? Well, we know from our previous theorem that is algebraic and so every element is algebraic including , , and . Thus, putting it all together we have the following:
Theorem: Let be an extension. Then, the algebraic closure of in is a subextension.
 Morandi, Patrick. Field and Galois Theory. New York: Springer, 1996. Print.
 Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
 Lang, Serge. Algebra. New York: Springer, 2002. Print.
 Conrad, Keith. Collected Notes on Field and Galois Theory. Web. <http://www.math.uconn.edu/~kconrad/blurbs/>.
 Clark, Pete. Field Theory. Web. <http://math.uga.edu/~pete/FieldTheory.p
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