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, \( \mathcal O_{L_D}/\frak P_D=\mathcal O_K/\frak p\).

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, \( x-\tau(\overline{\theta})=x-\overline{\sigma(\theta)}\) for some \( \sigma\in D\), so \( \tau(\overline{\theta})=\overline{\sigma(\theta)}\). \( \square\)

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. \( f(\frak P/\frak P_E)=1,e(\frak P/\frak P_E)=e\).

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.

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

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 \( e(\frak P’_D/\frak p)=1\).