Totally ramified extensions

Today we consider the case of a totally ramified extension of local fields , with residue fields —recall that this means . It turns out that there is a similar characterization as for unramified extensions.

 

Unramified extensions

As is likely the case with many math bloggers, I’ve been looking quite a bit at MO and haven’t updated on some of the previous series in a while.

Back to ANT. Today, we tackle the case . We work in the local case where all our DVRs are complete, and all our residue fields are perfect (e.g. finite) (EDIT: I don’t think this works out in the non-local case). I’ll just state these assumptions at the outset. Then, unramified extensions can be described fairly explicitly.

 

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.

 

Hensel's lemma and a classification theorem

So, I’ll discuss the proof of a classification theorem that DVRs are often power series rings, using Hensel’s lemma. 

Systems of representatives  

Let be a complete DVR with maximal ideal and quotient field . We let ; this is the residue field and is, e.g., the integers mod for the -adic integers (I will discuss this more later).

The main result that we have today is:

Theorem 1 Suppose is of characteristic zero. Then , the power series ring in one variable, with respect to the usual discrete valuation on .

 

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.  

 

DVRs II

Earlier I went over the definition and first properties of a discrete valuation ring.  Today, it’s time to say how we can tell a ring is a DVR–it turns out to be not too bad, which is nice because the properties we need in this criterion are often easier to work with than the existence of some discrete valuation.

Today’s result is:

Theorem 1 If the domain is Noetherian, integrally closed, and has a unique nonzero prime ideal , then is a DVR. Conversely, any DVR has those properties.

 

Discrete valuation rings and absolute values

I was initially planning on doing a post on Hensel’s lemma. Actually, I think I’ll leave that for later, after I’ve covered some more number theory (which may motivate it better).

So the goal for the next several posts is to cover some algebraic number theory, eventually leading into class field theory. At least in the near future, I intend to keep everything purely local. Thus, the appropriate place to start is to discuss discrete valuation rings rather than Dedekind domains. 

Absolute Values  

Actually, it is perhaps more logical to introduce discrete valuations as a special case of absolute values, which in turn generalize the standard absolute value on .