We shall establish relations between degrees, inertia degrees and ramification indices involved in decomposition and inertia field of a given prime in Galois extension. This is based on Marcus’s *Number Fields* and online notes by R. Ash. It follows a very similar appoach to, but is not based on, the one which can be found in this blog post by Sander Mack-Crane. For this blog post, understanding of Galois theory and basic facts about number fields is necessary.

First we fix some notation. Let be a Galois extension of number fields with Galois group and degree . For a subgroup we denote by its fixed field For any intermediate field we let be its ring of integers. If is a prime lying over in some field extension , we denote by the corresponding ramification index (i.e. the exponent of in factorization of ) and by the inertia degree (i.e. the degree of finite field extension ). We have the following facts, which we don’t prove here:

**Theorem 1:** If is a prime in and are two primes in lying over , then for some we have , hence . Denoting these common values by we moreover have , where is the number of distinct primes in lying over .

**Theorem 2:** and are multiplicative in towers, i.e. for a (not necessarily Galois) extensions tower and primes in respective integer rings, then .

From now on, fix a prime in lying under a prime in . For a subgroup we denote by the prime of lying between and . We focus our attention on two subgroups of :

**Definition:** We define the *decomposition group* of to be the set of preserving (i.e. ). We also define the *inertia group* of to be the set of preserving each congruence class modulo (i.e. ).

It’s clear that is a subgroup of and is a subgroup of . Our goal is to determine, for any two of the fields the degree of the extension, ramification index and inertia degree

It follows from theorem 1 that acts transitively on the set of all primes lying over , and can be seen as the stabilizer of for this action. By orbit-stabilizer theorem, so, again by theorem 1, .

Since is clearly Galois, we can apply theorem 1 to it as well to get for , where there is no term, since fixes , so there can be no other prime lying over . But clearly (e.g. because of multiplicativity in towers), so . Thus , so, in particular, $latex \mathcal O_{L_D}/\frak P_D=\mathcal O_K/\frak p$.

Since every element of preserves , we can view as permuting the congruence classes modulo , i.e. as a permutation of . Indeed, it’s clearly seen to be an automorphism of . Moreover, since fixes pointwise, fixes , so we have a natural homomorphism from to . From definition we see that is precisely the kernel of this homomorphism. So we have an injective homomorphism from into .

**Proposition:** The above homomorphism is an isomorphism.

**Proof:** We only need to show that it’s surjective. We will show that for any there is a such that . For that, let be such that is a generator of multiplicative group of . Then any automorphism in is determined by its value at . So we only need to find such that .

Let be the minimal polynomial of over and let be the minimal polynomial of over . Then , so . Since , for all . But , product running over (not necessarily all) , so . In particular, $latex x-\tau(\overline{\theta})=x-\overline{\sigma(\theta)}$ for some , so $latex \tau(\overline{\theta})=\overline{\sigma(\theta)}$.

Hence . Therefore, , so .

As , every element of fixes , so there can’t be any more primes lying over , since is Galois. By theorem 1 we have . But every automorphism in acts trivially on . But from the proposition maps onto the automorphism group of . Therefore the automorphism group, and hence this extension, must be trivial, i.e. $latex f(\frak P/\frak P_E)=1,e(\frak P/\frak P_E)=e$.

At this point, by repeatedly using multiplicativity in towers, we can easily get the following result.

**Theorem:** Let and as in theorem 1.

- , , ,
- , , ,
- , , .

**Remark:** Note that although primes lying over behave in many ways the same thanks to theorem 1, the definition of does depend on which we choose. Hence, for example, it needn’t be true that for some other prime in lying over we have .

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