My goal was to make this exposition as self-contained as possible. I only assumed that you knew about some basic concepts: polynomial rings, irreducibility, etc... Because of this, I require the statement of many theorems and must prove each one with some detail.
The part of this proof that should be most familiar to you is part one: showing that is a unique factorization domain. If not, I strongly recommend that you go back and examine the proof of the Fundamental Theorem of Arithmetic. You will find that this is a direct analogue. The "hard" part is really part two: showing that a polynomial in can be factored if and only if it can be factored in .