Let have degree and let have degree at least 1. Prove that if the leading coefficient of B is a unit of R, then there exist unique polynomials such that for all i and .
I'm a bit threw off by this problem, if B is a unit of R, what would that do?
Let have degree and let have degree at least 1. Prove that if the leading coefficient of B is a unit of R, then there exist unique polynomials such that for all i and .
I'm a bit threw off by this problem, if B is a unit of R, what would that do?
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 is why? thus since is a unit, we'll get which
contradicts thus:
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.
Here is another way to prove existence, though it is not general. It assumes that is a field (and so the division algorithm holds). But it gives us an algorithmic way to find the expansion for our favorite rings such as . It also seems to parallel the general way of expressing numbers in bases for positive integers.
Let with some fixed polynomial with .
Find the smallest such that and .
By division algorithm we get, .
Notice that and .
By division algorithm we get, .
Notice that and .
Continuing in this manner we get, where and .
Set .
It follows that,
.