e, f, and the remainder theorem

So, now to the next topic in introductory algebraic number theory: ramification. This is a measure of how primes “split.”  (No, definitely wrong word there…)

e and f 

Fix a Dedekind domain with quotient field ; let be a finite separable extension of , and the integral closure of in . We know that is a Dedekind domain.

(By the way, I’m now assuming that readers have been following the past few posts or so on these topics.)

Given a prime , there is a prime lying above . I hinted at the proof in the previous post, but to save time and avoid too much redundancy I’ll refer interested readers to this post.

Now, we can do a prime factorization of say . The primes contain and consequently lie above . Conversely, any prime of containing must lie above , since if is an ideal in a Dedekind domain contained in a prime ideal , then occurs in the prime factorization of (to see this, localize and work in a DVR).

 

Extensions of discrete valuations

With the school year starting, I can’t keep up with the one-post-a-day frequency anymore. Still, I want to keep plowing ahead towards class field theory.

Today’s main goal is to show that under certain conditions, we can always extend valuations to bigger fields. I’m not aiming for maximum generality here though. 

Dedekind Domains and Extensions  

One of the reasons Dedekind domains are so important is

Theorem 1 Let be a Dedekind domain with quotient field , a finite separable extension of , and the integral closure of in . Then is Dedekind.

 

Dedekind domains

Today’s (quick) topic focuses on Dedekind domains. These come up when you take the ring of integers in any finite extension of (i.e. number fields). In these, you don’t necessarily have unique factorization. But you do have something close, which makes these crucial. 

Definition 1 A Dedekind domain is a Noetherian integral domain that is integrally closed, and of Krull dimension one—that is, each nonzero prime ideal is maximal.