Normalization is finite, surjective and birational

Given: AC, the algebraically closed field $k$, the irreducible affine variety $X$ with coordinate ring $A=k[X]$ and function field $k(X)$, the integral closure $B$ of $A$ in $k(X)$, the normalization $X^{\nu }$ with $k[X^{\nu }]=B$ and the morphism $\nu$ whose pullback is the inclusion $A\hookrightarrow B$.

The normalization is finite and birational: $\nu$ is a finite morphism, and its pullback ${\nu }^{\ast }\text{\hspace{0.17em}⁣}:k(X)\to k(X^{\nu })$ is an isomorphism of $k$-extensions (The normalization of an irreducible affine variety, The normalization of an irreducible affine variety is finite, Irreducible affine varieties are birational exactly when their function fields are isomorphic, Birational maps and birational equivalence of classical affine varieties).

Lying over: if $A\to B$ is integral and $\mathfrak{p}\subseteq A$ is prime with $\ker \subseteq \mathfrak{p}$, there is a prime $\mathfrak{q}\subseteq B$ contracting to $\mathfrak{p}$; moreover a prime of $B$ is maximal exactly when its contraction to $A$ is (Lying over for integral ring maps, Under an integral extension, a prime is maximal if and only if its contraction is maximal). AC is used here.

Points of the affine varieties $X$ and $X^{\nu }$ correspond bijectively to maximal ideals of $A$ and $B$, and pullback of functions is the ring map induced by $\nu$, so the maximal ideal of $\nu (y)$ is the contraction of the maximal ideal of $y$ (Points of an affine algebraic set correspond to maximal ideals of its coordinate ring, The normalization of an irreducible affine variety).

Dominant morphisms pull back function fields, which is how the function-field isomorphism of [F1] is read as birationality of $\nu$ (Dominant maps pull back function fields functorially, Dominant morphisms and dominant rational maps); normality of $X^{\nu }$ is recorded in The normalization of an irreducible affine variety and Normality is checked on affine open charts.