Corollary 2

Let A be an integrally closed domain, K its field of fractions, K' a finite algebraic extension of K and A' a subring of K' containing A and integral over A.

(i) For every prime ideal $\mathfrak{p}$ of $A$ the set of prime ideals of $A'$ lying above $\mathfrak{p}$ is finite. (ii) If $\mathfrak{p}'$ is a prime ideal of $A'$ lying above $\mathfrak{p}$, every element of $A'/\mathfrak{p}'$ is of degree $\leq [\mathbf{K}':\mathbf{K}]$ over the field of fractions of $A/\mathfrak{p}$.

(i) Let $K''$ be the quasi-Galois extension of $K$ generated by $K'$ in an algebraic closure of $K'$ and $A''$ the integral closure of $A$ in $K''$. The field $K''$ is a finite extension of $K$ (Algebra, Chapter V, § 6, no. 3, Corollary 1 to Proposition 9) and hence its group of $K$-automorphisms is finite; it follows that the set of prime ideals of $A''$ lying above $\mathfrak{p}$ is finite (Proposition 6 (i)). On the other hand, as $A''$ is integral over $A$, the mapping $\mathfrak{p}'' \mapsto \mathfrak{p}'' \cap A'$ of the set of prime ideals of $A''$ lying above $\mathfrak{p}$ to the set of prime ideals of $A'$ lying above $x$ is surjective (no. 1, Theorem 1).

(ii) The coefficients of the minimal polynomial (over $K$) of any element $x' \in A'$ belong to $A$ ($§ 1$, no. 3, Corollary to Proposition 10); applying the canonical homomorphism $\pi': A' \to A'/\mathfrak{p}'$ to the coefficients of this polynomial, an equation of integral dependence with coefficients in $A/\mathfrak{p}$ and of degree $\leq [\mathbf{K}':\mathbf{K}]$ is obtained for the class mod. $\mathfrak{p}'$ of $x$; whence the conclusion.