For polynomials in one variable I know the proof:
Base case: All degree 1 polynomials are irreducible
Inductive step: Assume all polynomials of degree $\leq n$ can be factored into irreducibles. Then a polynomial of degree $n + 1$ or smaller is either irreducible or is the product of two factors whose degree are both $\leq n$, and hence are products of irreducibles.
Now how do I extend this to polynomials in multiple variables?
Can't I just replace "degree" with "total degree"?
My professor said you need to "pick a monomial ordering". He said you could also use induction on the number of variables. I don't understand either of those explanations.