Suppose is a number field that is Galois over
with
group
.
Fix a prime
lying over
.

**Definition 9.3.1** (Decomposition group)
The

*decomposition group* of

is the subgroup

Note that
is the stabilizer of
for
the action of on the set of primes lying over .
It also makes sense to define decomposition groups for relative
extensions , but for simplicity and to fix ideas in this section
we only define decomposition groups for a Galois extension
.

Let
denote the residue class field of
.
In this section we will prove that there is an exact sequence

where
is the *inertia subgroup* of
, and
, where is the exponent of
in the factorization of . The most interesting part of the proof is
showing that the natural map
is surjective.
We will also discuss the structure of
and introduce
Frobenius elements, which play a crucial role in understanding Galois
representations.

Recall from Theorem 9.2.2
that acts transitively on the set of primes
lying
over . The orbit-stabilizer theorem implies that
equals the cardinality of the
orbit of
, which by Theorem 9.2.2
equals the number of primes lying over , so
.

**Lemma 9.3.2**
*
The decomposition subgroups
corresponding to primes
lying over a given are all conjugate as subgroups of .*
*Proof*.
We have for each

, that

so

Thus

Thus

.

The decomposition group is useful because it allows us
to refine the extension
into a tower of extensions, such that at
each step in the tower we understand well the splitting behavior
of the primes lying over .

We characterize the fixed field of
as follows.

*Proof*.
First suppose

, and note that by Galois theory

, and by Theorem

9.2.2, the group

acts transitively on the primes of

lying over

. One of
these primes is

, and

fixes

by definition, so there is
only one prime of

lying over

, i.e.,

.
Conversely, if

is such that

has

, then

fixes

(since it is the only
prime over

), so

, hence

.

Thus does not split in going from to --it does some
combination of ramifying and staying inert. To fill in more of
the picture, the following proposition asserts that splits
completely and does not ramify in
.

**Proposition 9.3.4**
*
Fix a finite Galois extension of
,
let
be a prime lying over with decomposition group ,
and set .
Let
be for
and .
Then ,
,
and
.*
*Proof*.
As mentioned right after Definition

9.3.1, the
orbit-stabilizer theorem implies that

, and
by Galois theory

, so

.
Proposition

9.3.3,,

so
by Theorem

9.2.2,

Now

and

, so
we must have

and

.
Since

and

,
it follows that

.

**Subsections**
William Stein
2012-09-24