this result is an interesting generalization of the representation of polynomials. it looks so natural that i don't know why i've never thought about it myself?! so i'm going to have to thank you

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 iswhy?thus since is a unit, we'll get which

contradicts thus:

existence of the representation: we only need to prove the claim forwhy?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.