Corollary 3

Let $E$ be an extension of degree $n$ of the field $Q$ of rational numbers. The additive group of the integral closure in $E$ of the ring $Z$ of rational integers is a free commutative group of rank $n$.

$Z$ is integrally closed (no. 3, Proposition 10) and $E$ is separable since $Q$ is of characteristic 0. Corollary 2 can therefore be applied to the case where $A = Z$, $K = Q$ and $K' = E$.

Remark (2). The conclusions of Corollary 1 are not necessarily true if $K'$ is not assumed to be separate over $K$, even if $K'$ is an extension field of $K$ (Exercise 20). On the other hand, if $A$ is a finitely generated integral $K_0$-algebra, where $K_0$ is a field, the integral closure of $A$ in any extension of finite degree of the field of fractions of $A$ is a finitely generated $A$-module and a Noetherian ring, as we shall see in § 3, no. 2, Theorem 2.