Can we find an irreducible polynomial of any degree in $\mathbb C[x,y]$?

Question

Let's say that we have the following problem:

Given an integer $n \geq 1$, find an irreducible polynomial $p(x,y) \in \mathbb C[x,y]$ such that $\deg p = n$.

I've heard the fact that there is an irreducible polynomial of every non-zero degree a couple of times. I'm interested in a short proof of that fact or a reference.

Also, is there a nice family of irreducible polynomials $p_n \in \mathbb C[x,y]$ such that $\deg p_n = n$?

Answer 1

What about $p_n(x,y) = x + y^n$ ?