Affine space is normal

Given: AC, the algebraically closed field $k$, the integer $n\ge 0$, affine $n$-space $X={\mathbf{A}}_k^n$ with coordinate ring $A=k[x_1,\dots ,x_n]$, and a point $x\in X$ with maximal ideal ${\mathfrak{m}}_x\subseteq A$.

$A=k[x_1,\dots ,x_n]$ is an integrally closed domain, and it is Noetherian, so it is a normal Noetherian ring in the sense of normal noetherian ring (Finite-variable polynomial algebras over fields are integrally closed, Every algebra of finite type over a Noetherian ring is a Noetherian ring, A classical affine variety).

The local ring of $X$ at $x$ is the localisation ${\mathcal{O}}_{X,x}\cong A_{{\mathfrak{m}}_x}$ at the corresponding maximal ideal (The local ring at a point of an affine variety is the localization at its maximal ideal).

A domain is integrally closed if and only if all of its maximal localisations are integrally closed (A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are); the Noetherian ring $A$ is normal in the sense of normal noetherian ring precisely when its prime localisations are integrally closed domains.

$X$ is normal exactly when every local ring ${\mathcal{O}}_{X,x}$ is an integrally closed domain (Normal points and normal varieties).

AC is inherited through the classical localization, normalization, or finite-morphism suppliers cited above (The Axiom of Choice).