1) Division, you said it! Use the fact that .
is invertible in so such that:
, and i.e. for two unique . (can be written bette, when I write the identity refers to a whose coefficients are in so it's equal to a unique polynomial in and there is no problem, same thing fo and )
2) is quite easy! I don't think you really need 1)
3) Here 1) can be useful. You've guessed it, but how to prove that? Take a assume it belongs to and write it with the identity from 1). Your goal is to prove that
4)Take a . Is in ?
5)Suppose is reducible, i.e. .
Then and since is irreducible in which is factorial we have and for an invertible and a . This makes impossible, contradiction.
Therefore is irreducible in and you go on as you said.