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).
In particular, only finitely many primes of lie above a given prime of
.
Definition 1 If
lies above
, we write
for the number of times
occurs in the prime factorization of
. We call this the ramification index.We let
be the degree of the field extension
. This is called the residue class degree.
The ramification index has an interpretation in terms of discrete valuations. Let
be the absolute value on
corresponding to the prime
and, by abuse of notation,
its extension to
corresponding to
. Then
This is because if is a uniformizer of
(so that
generates the cyclic group
), and
at
, then
is a unit with respect to the absolute value
, or in the discrete valuation ring
.
A basic fact about and
is that they are multiplicative in towers; that is, if
is a finite separable extension,
the integral closure in
,
a prime lying over
which lies over
, we have:
The assertion about follows from (1), and that about
by the multiplicativity of degrees of field extensions. The degree
is also multiplicative in towers for the same reason. There is a similarity.
Proposition 2 For
, we have
Indeed, we may replace with
and
with
, where
. Localization preserves integral closure, and the localization of a Dedekind domain is one too (unless it is a field). Finally,
and
are stable under localization, which follows from the definitions.
In this case, is assumed to be a DVR, hence a PID. Thus
is a torsion-free, hence free, finitely generated
-module. Since
is free over
of rank
, the rank of
over
is
too. Thus
is a vector space over
of rank
. I claim that this rank is also
.
Indeed, let the factorization be . Now we have for
,
, so taking high powers yields
.
By the remainder theorem below, we have
as rings. We need to compute the dimension of each factor as an -vector space. Now we have
and since is principal (see below) all the successive quotients are isomorphic to
, which has dimension
. So counting dimensions gives the proof.
The remainder theorem and a consequence
The remainder theorem below was known for the integers for thousands of years, but its modern form is elegant.
Theorem 3 (Chinese Remainder Theorem) Let
be a ring and
be ideals with
for
. Then the homomorphism
is surjective with kernel
.
First we tackle surjectivity.
If the assertion is trivial. If
, then say
. We then have
where
. So, if we fix
and want to choose
with
the natural choice is .
For higher , it will be sufficient to prove that each vector with all zeros except for one 1 in the product occurs in the image. By symmetry, we need only show that there is
such that
while
for
.
Since for
,
i.e. . Now apply the
assertion to
to find
satisfying the conditions.
Now for the kernel assertion: we must show . For two ideals
it follows because one chooses
as above, and notes that
Then one uses induction and the fact that as above, etc.
As a corollary, we get a criterion for when a Dedekind domain is principal:
Theorem 4 A Dedekind domain
with finitely many prime ideals is principal.
To do this, we need only show that each prime is principal. Let the (nonzero) primes be ; we show
is principal. Choose a uniformizer
at
, i.e.
. Now choose
such that
Then and
have equal orders at all primes, hence are equal.
We tacitly used this theorem above: there has only finitely many prime ideals, which are the localizations of
, so is principal.
October 6, 2009 at 6:42 pm
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