The Degree of an Extension (Pt. II)
Point of Post: This is a continuation of this post.
One of the interesting things about extensions is that they transfer well. Namely, suppose that we have some extension and some subextension . A natural question is if there is some relation between the three degrees , , and that are floating around. Somewhat surprisingly, not only is there some connection between them, but it’s about as nice as one can hope:
Theorem: Let be an extension and a subextension. Then,
In fact, let and be bases for and respectively. Then, is a basis for which has cardinality .
Proof: Let’s first prove that really is a basis for . To prove that is linearly dependent we merely suppose that
with , all but finitely many zero. We see then that we may rewrite this as
Now, note that each inner sum lies in and since the are independent over we may conclude that the inner sum is zero for each . Now, fix a , the fact that
and that the are independent over tells us that each is zero, and since this was for an arbitrary we may conclude that all the are zero. To see that spans over we merely note that if is arbitrary then (by virtue of the fact that is a basis for ) there exists for each such that
But, since is a basis for there exists such that
The only order of business left to take care of is to show that which, if there is any justice in the world, should be true because the natural surjection is an injection. To see this suppose that . Since the ‘s are a basis for this implies that either or and . Clearly the first can’t happen since is a basis for is a basis and thus can’t contain zero, and so the second must be true. Now, we know that and since the distinctly indexed ‘s are distinct we may conclude that as well. The conclusion follows.
From this we get two immediate, important corollaries:
Corollary: Let be an extension and a subextension. Then, is finite if and only if and are.
Corollary: Let be a finite extension. Then, if is a subextension then and divide
The first corollary is often times useful for allowing us to prove a certain extension is finite by breaking it into a sequence of subextensions and treating each piece separately.
The second corollary is surprisingly powerful. For example, it seems kind of obvious that can’t be written as a rational polynomial in , right? Go ahead and try to prove this. In fact, it’s not extremely easy from standard (non-clever) means, but the second corollary allows us to discount this possibility immediately. Indeed, if this were true then which will tell us that , but this can’t be true since (as we shall soon verify) and and .
Perhaps even cooler is the sort of amazing fact that if is a finite extension of degree and then there can’t be a field with –in particular, if is prime then the only fields with is or .
Our last note in this post shall be one on notation. We have been having to say “Let be an extension and a subextension”, which is a lot of words. A more preferable notation attached to the notion of a tower of fields. In particular, we shall call a descending chain of fields a tower and denote it . The tower term comes from the other way to visually represent a tower which is by the Hasse diagram of the fields, with respect to inclusion. For example, would be written
Often times the degree of the extension is written beside the line, so that if then we’d write
 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.pdf>.