Totally ramified extensions October 23, 2009Posted by Akhil Mathew in algebra, algebraic number theory, number theory.
Tags: discrete valuation rings, Eisenstein polynomials, ramification, 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.
So, hypotheses as above, let be the rings of integers and choose a uniformizer of the DVR . I claim that . This follows easily from:
Lemma 1 Let be a DVR and let be a set of elements of whose image under reduction contains each element of the residue field of . Let be a uniformizer of . Then each can be written
where each .
This is a basic fact, which I discussed earlier, about systems of representatives.
Now, note that as a consequence. Consider the minimal polynomial of ,
Because of the total ramification hypothesis, any two terms in the above sum must have different orders—except potentially the first and the last. Consequently the first and the last must have the same orders in (that is, ) if the sum is to equal zero, so is a unit, or is a uniformizer in . Moreover, it follows that none of the , can be a unit—otherwise the order of the term would be , and the first such term would prevent the sum from being zero.
Note that I have repeatedly used the following fact: given a DVR and elements of pairwise distinct orders, the sum is nonzero.
In particular, what all this means is that is an Eisenstein polynomial:
Proposition 2 Given a totally ramified extension , we can take with and such that the irreducible monic polynomial for is an Eisenstein polynomial.
Now we prove the converse:
Proposition 3 If is an extension with where satisfies an Eisenstein polynomial, then and is totally ramified.
Note first of all that the Eisenstein polynomial mentioned in the statement is necessarily irreducible. As before, I claim that
is a DVR, which will establish one claim. By the same Nakayama-type argument in the previous post, one can show that any maximal ideal in contains the image of the maximal ideal ; in particular, it arises as the inverse image of an ideal in
this ideal must be . In particular, the unique maximal ideal of is generated by for a generator of . But, since is Eisenstein and the leading term is , it follows that . This also implies that is nonnilpotent in .
Now any commutative ring with a unique principal maximal ideal generated by a nonilpotent element is a DVR, and this implies that is a DVR. This is a lemma in Serre, but we can take a slightly quicker approach to prove this. We can always write a nonzero as for a unit, because for some —this is the Krull intersection theorem (Cor. 6 here). Thus from this representation is a domain, and the result is then clear.
Back to the proof of the second proposition. There is really only one more step, viz. to show that is totally ramified. But this is straightforward, because is Eisenstein, and if there was anything less than total ramification then one sees that would be nonzero—indeed, it would have the same order as the last constant coefficient .