I'm trying to use induction but I don't know how to apply the induction hypothesis.
Lemma. Let K be a finitely generated algebra over a field k. If K is a field, then K is a finite field extension of k.
Proof. We know $K=k[a_1,\dotsc,a_n]$, we can suppose $a_i \in K\setminus k$. We're going to see every $a_i$ is algebraic over $k$, because if $a_i$ is algebraic, $k[a_i]$ is a finite field extension. By induction, $K$ is a finite field extension.
We proceed by induction. If $n=1$, $K=k[a_1]$. For every $p\in K$ $\exists b_0,\dotsc,b_s \in k$ such that $$ p = \sum_{i=0}^s b_i a_1^i $$ Taking $p=a_1^{-1}$, it's clear that $a_1$ it's a root of $f(X) = -1+\sum_{i=0}^s b_i X^{i+1}$, so $a_1$ is algebraic over $k$.
Now I have to apply induction hypothesis. We $\exists f:{k[a_1,\dotsc,a_n]}\to{K}$ surjective, such that $K\cong k[a_1,\dotsc,a_n]/\ker(f)$. Obviusly, $\ker(f)$ is maximal ideal.
Can you give me a hint?