Corollary 2

Let $A'$ be an integral domain, $\mathscr{G}$ a finite group operating on $A'$, $A'$ the ring of invariants of $\mathscr{G}$ and $p'$ a prime ideal of $A'$; let $K, K^\mathbf{Z}$ and $K$ be the fields of fractions of $A, A^\mathbf{Z}$ and $A'$ respectively. Then $K'$ is a Galois extension Of $K$ and the subfields $L$ of $K$ containing $K$ and such that $p'$ is the only prime ideal of $A'$ lying above the ideal $p' \cap L$ of $A' \cap L$ are just those which contain $K^\mathbf{Z}$.

$\mathscr{G}$ operates on $K'$ and $K$ is the field of invariants of $\mathscr{G}$ in $K'$ (§ 1, no. 9, Proposition 23 applied to $S = A - \{0\}$) and similarly $K^\mathbf{Z}$ is the field of invariants of $\mathscr{G}^\mathbf{Z}$; by definition $K'$ is therefore a Galois extension of $K$. If is the subgroup of $\mathscr{G}$ consisting of those $\sigma \in \mathscr{G}$ leaving invariant the elements of L, to say that L contains $K^{\mathbf{Z}}$ means that $\mathscr{H}$ is contained in $\mathscr{G}^{\mathbf{Z}}$ (Algebra, Chapter V, § 10, no. 5, Theorem 3) and, as L is the field of invariants of $\mathscr{H}$ in $K'$, $A' \cap L$ is the ring of invariants of $\mathscr{H}$ in $A'$; the second assertion then follows from Proposition 4 (i).