Let $K$ be an algebraically closed field. Let $n$ be an integer. Let $M$ be a maximal ideal of the ring of polynomial $K[X_1,...,X_n]$. Then the quotient ring $K[X_1,...,X_n]/M$ a field and a $K$-algebra of finite type. So by the Lemma of Zariski, the field $K[X_1,...,X_n]/M$ is a finite extension of $K$. $K$ being algebrically closed, $K[X_1,...,X_n]/M$ is isomorphic to $K$.
My question is: does there exist an isomorphism of fields $\sigma$ from $K[X_1,...,X_n]/M$ to $K$ such that for all $a\in K$, the image of the equivalent class of $a$ in $K[X_1,...,X_n]/M$ by $\sigma$ is $a$ itself?
The theorem of zero implies that there exists $a_1,...,a_n$ such that $M$ is the ideal generated by $X-a_1,...,X-a_n$ consider $f(X_i)=a_i$ on $K[X_1,...,X_n]$. Remark that $f(P)=P(a_1,..,a_n)$. In particular, if $P$ is the constant polynomial $a$, $f(P)=a$. The morphism $f$ induces $f':K[X_1,...,X_n]/M$ such that $f'(a)=a$.