for sharing it with us!
uniqueness of the representation: since the leading coefficient of is a unit, for any we have now suppose with
let and be the leading coefficients of and repectively. then the leading coefficient of is why? thus since is a unit, we'll get which
existence of the representation: we only need to prove the claim for why? proof by induction over : if n = 0, then A = 1 and we put then so suppose the claim holds
for any with k < n. if then just choose and so we may assume that let since is a unit, we'll have:
which gives us: now apply the induction hypothesis to each term in the RHS
to finish the proof.