Reference request for a proof of a basic version of normalization lemma

I was reading some notes on affine algebraic geometry and came across the following fact:

If $k$ is a field with $\mathrm{char}~k = 0$, then if $V(f)$ is an irreducible hypersurface in $\mathbf{A}^n_k$, then there is a surjective linear projection $\pi: \mathbf{A}^n_k \to \mathbf{A}^{n-1}_k$ such that the composition $V(f) \to \mathbf{A}^n_k \to \mathbf{A}^{n-1}_k$ is finite.

Can someone point me to resource (book/lecture notes) where I can find this statement proved?

• I think you are asking about the Noether's normalization theorem (geometric version). This is an standard fact, thus you can find it easily in algebra (e.g. Atiyah-MacDonald) or algebraic geometry textbooks (e.g. Shafarevich). I hope it helps. Since this is a particular case of a general and important theorem, I think it's worth trying a proof by yourself. – user90189 Feb 24 '18 at 23:22