Proposition 7

*Let $A$ be an integrally closed domain, $K$ its field of fractions, $K'$ a Galois extension of $K$, $\mathscr{G}$ its Galois group, $A'$ the integral closure of $A$ in $K'$, $p'$ a prime ideal of $A'$ and $p = A \cap p'$. Finally, let $L$ be a subfield of $K'$ containing $K$ and let $p(L) = p' \cap L$.

(i) The decomposition field (resp. inertia field) *of $p'$ with respect to $L$ is $L(K^{\mathbf{Z}})$ (resp. $L(K^T)$); *if further $L$ is a Galois extension of $K$, the decomposition field of $p(L)$ with respect to $K$ is $L \cap K^{\mathbf{Z}}$.

(ii) *If $L$ is contained in $K^{\mathbf{Z}}$, $A/p$ and $(A' \cap L)/p(L)$ have the same field of fractions and in the local ring $A' \cap L$ corresponding to the prime ideal $p(L)$, the maximal ideal is generated by $p$. Conversely, if these two conditions hold and further $\bigcap_{n > 0} p^n A'_p = 0$, $L$ is contained in $K^{\mathbf{Z}}$.

(i) If $\mathscr{H}$ is the subgroup of $\mathscr{G}$ leaving invariant the elements of $L$, clearly the decomposition group (resp. inertia group) of $p'$ with respect to $L$ is $G^{\mathbf{Z}} \cap \mathscr{H}$ (resp. $G^T \cap \mathscr{H}$) and the first assertion follows from Galois theory if $K'$ is a finite Galois extension of $K$ (Algebra, Chapter V, § 10, no. 6, Corollary 1 to Theorem 3); in the general case it follows from the fact that $A^{\mathbf{Z}}$ (resp. $A^T$) is the union of the $A^{\mathbf{Z}}(p' \cap K_\alpha)$ (resp. $A^T(p' \cap K_\beta)$) in the notation of Remark 2: every element $x \in K'$ belongs to some $K$, and if it is invariant under $G^{\mathbf{Z}}(p') \cap \mathscr{H}$ (resp. $G^T(p') \cap \mathscr{H}$) it is also invariant under $G^{\mathbf{Z}}(p' \cap K_{\beta'}) \cap \mathscr{H}$ (resp. $G^T(p' \cap K_{\beta'}) \cap \mathscr{H}$) for some suitable $\beta'$; hence it belongs by the beginning of the argument to

Suppose now that $L$ is a Galois extension of $K$; the restriction to $L$ of every $\sigma \in G^{\mathbf{Z}}$ then leaves invariant $p(L) = p' \cap L$ and hence belongs to the decomposition group of $p(L)$ with respect to $K$. Conversely, let $\tau$ be an automorphism of $L$ belonging to this group and let $\sigma$ be an extension of $\tau$ to a $K$-automorphism of $K'$; we write $q' = \sigma.p'$. As $p'$ and $q'$ both lie above $p(L)$, there exists an automorphism $\rho \in \mathscr{H}$ such that $q' = \rho.p'$, whence $\rho^{-1}\sigma \in G^{\mathbf{Z}}$ and $\tau$ is the restriction of $\rho^{-1}\sigma$ to $L$; in other words, the decomposition group of $p(L)$ with respect to $K$ is identical with the group of restrictions to $L$ of the automorphisms $\sigma \in G^{\mathbf{Z}}$, which proves the second assertion.

(ii) To say that $L \subset K^{\mathbf{Z}}$ means that $\mathscr{H} \supset G^{\mathbf{Z}}$ and the assertions of (ii) are therefore special cases of no. 2, Proposition 4 (ii) and (iii) when $[K':K]$ is finite. In the general case the argument is as in the proof of Proposition 6.