deepdream of a sidewalk
Show Answer
Math and science::Algebra::Aluffi

Rings. Uniqueness of polynomial division.

Two versions of the same idea: unique polynomial division.

Let R be a ring, R[x] be a polynomial ring and let f(x),g(x)∈R[x] be polynomials, with f(x) being a monic polynomial.

Then there exists a unique pair of polynomials q(x),r(x)∈R[x] where degr<degf such that:

g(x)=q(x)f(x)+r(x)

In terms of cosets:

Let R be a ring, R[x] be a polynomial ring and let f(x),g(x)∈R[x] be polynomials, with f(x) being a monic polynomial.

Then there exists a unique polynomial r(x)∈R[x] where degr<degf such that:

[ g(x)+?=r(x)+?]

These theorems rely on a lemma showing polynomial long division to be unique when f(x) is monic. Can you remember the proof?