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
.
October 23, 2009 at 8:59 pm
This is a little off-topic, but congratulations on Siemens semi!
I’m also curious what sources you’re working out of in these posts. I’ve been looking for a text that approaches algebraic number theory from a more commutative algebra point of view.
October 23, 2009 at 9:14 pm
Thanks!
In fact I’m looking for a book that approaches ANT from a more number-theoretic point of view- I’ve gotten more comfortable with the general theory than when I first started, but I was under the impression that most sources tended to entirely ignore examples (to my disappointment).
The sources that I’m primarily using are Serre’s Local Fields, Lang’s Algebraic Number Theory, and Cassels-Frohlich. All these are heavily commutative-algebraic (especially the first two). I’ve also found the first few chapters of Weil’s Basic Number Theory useful, but the writing is somewhat dense for me and I have not gotten that far.
October 23, 2009 at 9:50 pm
Stein has a great deal of examples and I think it complements a heavy-duty approach very well.
October 24, 2009 at 6:13 pm
For some reason both your comments got caught in the spam filter (presumably you posted the second one when the first didn’t appear).
Thanks for the link!
October 6, 2010 at 1:10 pm
hey has anyone tried “Theory of Algebraic integers” Richard Dedikind. Pierre samuel’s book is also a good one to start of with.