Suppose
is a number field that is Galois over
with
group
.
Fix a prime
lying over
.
(Note: The decomposition group is called the ``splitting group''
in Swinnerton-Dyer. Everybody I know calls it the decomposition
group, so we will too.)
Let
denote the residue class field of
.
In this section we will prove that there is a natural exact sequence
where
is the of
, and
. 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 roll in understanding Galois
representations.
Recall that
acts on the set of primes
lying
over
. Thus the decomposition group is the stabilizer in
of
. The orbit-stabilizer theorem implies that
equals the orbit of
, which by Theorem 13.2.2
equals the number
of primes lying over
, so
.
Lemma 14.1.2
The decomposition subgroups
corresponding to primes
lying over a given
are all conjugate in
.
Proof.
We have
![$ \tau(\sigma(\tau^{-1}(\mathfrak{p}))) = \mathfrak{p}$](img1287.png)
if and only if
![$ \sigma(\tau^{-1}(\mathfrak{p})) = \tau^{-1}\mathfrak{p}$](img1288.png)
. Thus
![$ \tau\sigma\tau^{-1}\in D_p$](img1289.png)
if and only if
![$ \sigma\in D_{\tau^{-1}\mathfrak{p}}$](img1290.png)
, so
![$ \tau^{-1}D_p\tau = D_{\tau^{-1}\mathfrak{p}}$](img1291.png)
. The lemma now follows
because, by Theorem
13.2.2,
![$ G$](img31.png)
acts transitively on the set of
![$ \mathfrak{p}$](img357.png)
lying over
![$ p$](img4.png)
.
The decomposition group is extremely useful because it allows us
to see the extension
as a tower of extensions, such that at
each step in the tower we understand well the splitting behavior
of the primes lying over
. Now might be a good time to glance
ahead at Figure 14.1.2 on page
.
We characterize the fixed field of
as follows.
Proof.
First suppose
![$ L=K^D$](img1300.png)
, and note that by Galois theory
![$ \Gal (K/L)\cong
D$](img1301.png)
, and by Theorem
13.2.2, the group
![$ D$](img1294.png)
acts transitively on the primes of
![$ K$](img9.png)
lying over
![$ \mathfrak{p}\cap L$](img1298.png)
. One of
these primes is
![$ \mathfrak{p}$](img357.png)
, and
![$ D$](img1294.png)
fixes
![$ \mathfrak{p}$](img357.png)
by definition, so there is
only one prime of
![$ K$](img9.png)
lying over
![$ \mathfrak{p}\cap L$](img1298.png)
, i.e.,
![$ \mathfrak{p}\cap L$](img1298.png)
does not
split in
![$ K$](img9.png)
. Conversely, if
![$ L\subset K$](img1297.png)
is such that
![$ \mathfrak{p}\cap L$](img1298.png)
does not split in
![$ K$](img9.png)
, then
![$ \Gal (K/L)$](img1166.png)
fixes
![$ \mathfrak{p}$](img357.png)
(since it is the only
prime over
![$ \mathfrak{p}\cap L$](img1298.png)
), so
![$ \Gal (K/L)\subset D$](img1302.png)
, hence
![$ K^D\subset L$](img1303.png)
.
Thus
does not split in going from
to
--it does some
combination of ramifying and staying inert. To fill in more of
the picture, the following proposition asserts that
splits
completely and does not ramify in
.
Proposition 14.1.4
Let
for our fixed prime
and Galois extension
.
Let
be for
and
.
Then
and
, i.e.,
does not ramify and splits
completely in
. Also
and
.
Proof.
As mentioned right after Definition
14.1.1, the
orbit-stabilizer theorem implies that
![$ g(K/\mathbf{Q})=[G:D]$](img1310.png)
, and
by Galois theory
![$ [G:D]=[L:\mathbf{Q}]$](img1311.png)
.
Thus
Now
![$ e(K/L)\leq e(K/\mathbf{Q})$](img1315.png)
and
![$ f(K/L)\leq f(K/\mathbf{Q})$](img1316.png)
, so
we must have
![$ e(K/L)=e(K/\mathbf{Q})$](img1317.png)
and
![$ f(K/L)=f(K/\mathbf{Q})$](img1318.png)
.
Since
![$ e(K/\mathbf{Q})=e(K/L)\cdot e(L/\mathbf{Q})$](img1319.png)
and
![$ f(K/\mathbf{Q})=f(K/L)\cdot f(L/\mathbf{Q})$](img1320.png)
,
the proposition follows.
Subsections
William Stein
2004-05-06