Decomposition and inertia field degrees

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 L/K be a Galois extension of number fields with Galois group G and degree n. For a subgroup H\leq G we denote by L_H its fixed field For any intermediate field F we let \mathcal O_F be its ring of integers. If \frak P is a prime lying over \frak p in some field extension F_1/F_2, we denote by e(\frak P/\frak p) the corresponding ramification index (i.e. the exponent of \frak P in factorization of \frak p\mathcal O_{F_2}) and by f(\frak P/\frak p) the inertia degree (i.e. the degree of finite field extension [(\mathcal O_{F_2}/\frak P):(\mathcal O_{F_1}/\frak p)]). We have the following facts, which we don’t prove here:

Theorem 1: If \frak p is a prime in \mathcal O_K and \frak P_1,\frak P_2 are two primes in \mathcal O_L lying over \frak p, then for some \sigma\in G we have \frak P_2=\sigma(\frak P_1), hence e(\frak P_1/\frak p)=e(\frak P_2/\frak p),f(\frak P_1/\frak p)=f(\frak P_2/\frak p). Denoting these common values by e,f we moreover have efg=n, where g is the number of distinct primes in \mathcal O_L lying over \frak p.

Theorem 2: e and f are multiplicative in towers, i.e. for a (not necessarily Galois) extensions tower F_1/F_2/F_3 and primes \frak p_1\supseteq\frak p_2\supseteq p_3 in respective integer rings, then e(\frak p_3/\frak p_1)=e(\frak p_3/\frak p_2)e(\frak p_2/\frak p_1),f(\frak p_3/\frak p_1)=f(\frak p_3/\frak p_2)f(\frak p_2/\frak p_1).

From now on, fix a prime \frak p in \mathcal O_K lying under a prime \frak P in \mathcal O_L. For a subgroup H\leq G we denote by \frak P_H the prime of O_{L_H} lying between \frak p and \frak P. We focus our attention on two subgroups of G:

Definition: We define the decomposition group D of \frak P to be the set of \sigma\in G preserving \frak P (i.e. \sigma(\frak P)=\frak P). We also define the inertia group E of \frak P to be the set of \sigma\in G preserving each congruence class modulo \frak P (i.e. \sigma(\alpha)\equiv\alpha\pmod{\frak P}).

It’s clear that D is a subgroup of G and E is a subgroup of D. Our goal is to determine, for any two of the fields K,L_D,L_E,L the degree of the extension, ramification index and inertia degree

It follows from theorem 1 that G acts transitively on the set of all primes lying over \frak p, and D can be seen as the stabilizer of \frak P for this action. By orbit-stabilizer theorem, |G|=|D|g so, again by theorem 1, |D|g=efg,|D|=ef.

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

Since every element of D preserves \frak P, we can view \sigma\in D as permuting the congruence classes modulo \frak P, i.e. as a permutation \overline{\sigma} of \mathcal O_L/\frak P. Indeed, it’s clearly seen to be an automorphism of \mathcal O_L/\frak P. Moreover, since \sigma fixes K pointwise, \overline{\sigma} fixes \mathcal O_K/\frak p, so we have a natural homomorphism from D to \overline{G}=\mathrm{Gal}((\mathcal O_L/\frak P)/(\mathcal O_K/\frak p)). From definition we see that E is precisely the kernel of this homomorphism. So we have an injective homomorphism from D/E into \overline{G}.

Proposition: The above homomorphism is an isomorphism.

Proof: We only need to show that it’s surjective. We will show that for any \tau\in\overline G there is a \sigma\in D such that \overline\sigma=\tau. For that, let \theta\in \mathcal O_L be such that \overline\theta is a generator of multiplicative group of \mathcal O_L/\frak P. Then any automorphism in \overline G is determined by its value at \overline\theta. So we only need to find \sigma\in D such that \overline{\sigma(\theta)}=\tau(\overline\theta).
Let h(x) be the minimal polynomial of \theta over \mathcal O_{L_D} and let k(x) be the minimal polynomial of \overline\theta over \mathcal O_{L_D}/\frak P_D=\mathcal O_K/\frak P. Then \overline{h(\theta)}=\overline 0, so k(x)\mid\overline{h(x)}. Since k(x)=\prod_{\tau\in\overline G}(x-\tau(\overline\theta)), x-\tau(\overline\theta)\mid\overline{h(x)} for all \tau\in\overline G. But h(x)=\prod(x-\sigma(\theta)), product running over (not necessarily all) \sigma\in D, so (x-\tau(\overline\theta))\mid\overline{h(x)}=\prod(x-\overline{\sigma(\theta)}). In particular, $latex x-\tau(\overline{\theta})=x-\overline{\sigma(\theta)}$ for some \sigma\in D, so $latex \tau(\overline{\theta})=\overline{\sigma(\theta)}$. \square

Hence D/E\cong\overline G. Therefore, \frac{|D|}{|E|}=|D/E|=|\overline G|=f, so |E|=\frac{|D|}{f}=\frac{ef}{f}=e.

As E\subseteq D, every element of E fixes \frak P, so there can’t be any more primes lying over \frak P_E, since L/L_E is Galois. By theorem 1 we have e=|E|=e(\frak P/\frak P_E)f(\frak P/\frak P_E). But every automorphism in E acts trivially on \mathcal O_L/\frak P. But from the proposition E maps onto the automorphism group of (\mathcal O_L/\frak P)/(\mathcal O_{L_E}/\frak P_E). 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 e=e(\frak P/\frak p),f=(\frak P/\frak p) and g as in theorem 1.

  • [L:L_E]=ee(\frak P/\frak P_E)=e, f(\frak P/\frak P_E)=1,
  • [L_E:L_D]=fe(\frak P_E/\frak P_D)=1f(\frak P_E/\frak P_D)=f,
  • [L_D:K]=ge(\frak P_D/\frak p)=1f(\frak P_D/\frak p)=1.

Remark: Note that although primes lying over \frak p behave in many ways the same thanks to theorem 1, the definition of D,E does depend on which \frak P we choose. Hence, for example, it needn’t be true that for some other prime \frak P' in \mathcal O_L lying over \frak p we have e(\frak P'_D/\frak p)=1.


2 thoughts on “Decomposition and inertia field degrees

  1. Pingback: Higher ramification groups – Abstraction

  2. Pingback: Higher ramification groups – Abstraction

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