It's hard to give you advice because I don't know how much algebra you know or how much algebra Shekelberg expects you to know.
All factorization proofs go basically the same way: if a isn't irreducible, then you break a into a=p1p2 and apply the same recursion to p1 and p2.
The only nontrivial part of this argument is proving that you eventually hit an irreducible element this way, that is, you can't have an infinite chain such that p2 | p1, p3 | p2, p4 | p3... and never hit the end. Basically, you're proving it's Noetherian.
How easy this is depends on how much you know about rings of integers. If you're okay saying they're finitely generated rings, it's pretty easy. If you can't say that much, then I don't know how you would approach this without a sledgehammer from algebra.