I need to prove the following:

"Prove that is irreducible in if and only if is irreducible in "

I want to prove "if"; that is, irreducible irreducible.

So, I think we have that , with one of .

The hint states that we should argue by contradiction. So, in this instance, we shall assume that is reducible, and hence

where

Now,

and so,

we say, , and so

with , by the division algorithm. But, this is a contradiction, and so . Thus,

which contradicts the fact that is a unique factorisation domain.

I think this is wrong. I'd be really grateful if anyone could pick out the flaws in this proof, and if possible, provide a better (or, correct) proof.

Thanks in advance,

HTale.