For any Galois extension $L/K$ of algebraic number fields with rings of integers $\mathcal{O}_L$ and $\mathcal{O}_K$, any prime $\mathfrak{p} \subset \mathcal{O}_K$, and any prime $\mathfrak{P} \subset \mathcal{O}_L$ above $\mathfrak{p}$, one can define the decomposition subgroup of $\mathfrak{P}$ as: $$ D_{\mathfrak{P}}:=\{\varphi \in \textrm{Gal}(E/K) \mid \varphi(\mathfrak{P}) \}, $$ and Galois group of the corresponding residue field extension: $$ \textrm{Gal}((\mathcal{O}_L/\mathfrak{P})/(\mathcal{O}_K/\mathfrak{p})). $$ One can then define a homomorphism: $$D_{\mathfrak{P}} \to \textrm{Gal}((\mathcal{O}_L/\mathfrak{P})/(\mathcal{O}_K/\mathfrak{p})) \qquad \textrm{given by} \qquad \varphi \mapsto \widehat{\varphi}, $$ where $\widehat{\varphi}: \mathcal{O}_L/\mathfrak{P} \to \mathcal{O}_L/\mathfrak{P}$ is given by $\widehat{\varphi}(x\ (\textrm{mod}\ \mathfrak{P})) := \varphi(x)\ (\textrm{mod}\ \mathfrak{P})$ for all $x \in \mathcal{O}_L$.
Showing that this map is well-defined and that it constitutes a homomorphism is simply a matter of routine checking. However: It is said that the map is also surjective. I have been trying to prove this.
We want to show that every $(\mathcal{O}_K/\mathfrak{p})$-automorphism in $\textrm{Gal}((\mathcal{O}_L/\mathfrak{P})/(\mathcal{O}_K/\mathfrak{p}))$ is of the form $x\ (\textrm{mod}\ \mathfrak{P}) \mapsto \sigma(x)\ (\textrm{mod}\ \mathfrak{P})$, but I cannot seem to see why this should be so.
I have been trying to follow the proof given in chapter I, §9, p. 56 (proposition 9.6) of Algebraic Number Theory by J. Neukirch, but he proves it for the (more general) Dedekind domains, which seems to complicate matters.
Does anyone know an elegant proof of the surjectivity of the above homomorphism, or a reference to other textbook proofs?
Many thanks.
If you do not need results regarding infinite extensions of number fields (like $\bar{K}/K$, where $K$ is a number field and $\bar{K}$ is the algebraic closure), one of the best reference is Marcus' book Number Fields. There is a LaTeX second edition. In chapter 4 he develops the theory you need (of course, if you are not used to work with ramification, read also chapter 3). In particular, corollary 1 is the result you are looking for. The proof is not direct but is based on a cardinality argument, and based on the behaviour of the fixed field of inertia and decomposition subgroup.
Neukirch's book is really useful if you need to work with infinite extensions of local (henselian) fields. In this case, you need to work with valuations, and "Galois theory of valuations" is part of the second chapter of his book. But in the finite case, probably there's no need to bring up this theory.