Proposition 12

Let $\mathbf{A}$ be a Krull domain, $K$ its field of fractions, $K'$ a finite extension of $K$ and $\mathbf{A}'$ the integral closure of $\mathbf{A}$ in $K'$. Then $\mathbf{A}'$ is a Krull domain. The essential valuations of $\mathbf{A}'$ are the normed discrete valuations on $K'$ which are equivalent to the extensions of the essential valuations of $\mathbf{A}$.

Let $(v_i)_{i \in I}$ be the family of extensions to $K'$ of the essential valuations of $\mathbf{A}$. Since the degree $n = [K':K]$ is finite, the $v_i$ are discrete valuations on $K'$ (Chapter VI, § 8, no. 1, Corollary 3 to Proposition 1). Let $B_i$ be the ring of $v_i$; then $\mathbf{A}' \subset \bigcap_{L \in I} B_i$ (Chapter VI, § 1, no. 3, Theorem 3). Conversely, every element $x$ of $\bigcap_{i \in I} B_i$ is integral over each of the essential valuation rings of $\mathbf{A}$ (Chapter VI, § 1, no. 3, Corollary 3 to Theorem 3); hence the coefficients of the minimal polynomial of $x$ over $K$ belong to $\mathbf{A}$ (Chapter V, § 1, no. 3, Corollary to Proposition 11), so that $x \in \mathbf{A}'$; thus $\mathbf{A}' = \bigcap_{i \in I} B_i$. Now let $x$ be a non-zero element of $\mathbf{A}'$; it satisfies an equation of the form $x^s + a_{s-1}x^{s-1} + \ldots + a_0 = 0$

where $a_i \in \mathbf{A}$ and $a_s \neq 0$; if $v_i(x) > 0$, then $v_i(a_0) > 0$; now the essential valuations $v$ of $\mathbf{A}$ such that $v(a_0) > 0$ are finite in number and the valuations on $K'$ extending a given valuation on $K$ are also finite in number (Chapter VI, § 8, no. 3, Theorem 1); hence $v_i(x) = 0$ except for a finite number of indices $i \in I$. Thus it has been proved that $\mathbf{A}'$ is a Krull domain (no. 3, Definition 3).

It remains to show that the $v_i$ are equivalent to essential valuations of $A'$ (no. 4, Corollary 2 to Proposition 6), that is (no. 6, Corollary 2 to Theorem 3) that the prime ideal $p_i$, consisting of the $x \in A'$ such that $v_i(x) > 0$, is of height 1. If this were not so, there would exist a prime ideal $q$ of $A'$ such that $(0) \subset q \subset p_i$ distinct from $(0)$ and $p_i$; then $(0) \subset q \cap A \subset p_i \cap A$ and $q \cap A$ would be distinct from $(0)$ and $p_i \cap A$ (Chapter V, §2, no. 1, Corollary 1 to Proposition 1); the prime ideal $p_i \cap A$ would therefore not be of height 1, which contradicts the fact that it corresponds to an essential valuation of $A$.