# The Decomposition Group

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.

Proposition 9.3.3   The fixed field

for all

of is the smallest subfield such that the prime ideal has , i.e., there is a unique prime of over .

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