This is a generalization of the ordinary representation of polynomials:

**Problem**. Let be a commutative ring with and have degree and let have degree at least . Prove that if the leading coefficient of is a unit of , then there exist unique polynomials such that for all , and .

**Solution**. *Uniqueness of the representation *: Since the leading coefficient of is a unit, for any we have Now suppose that with Let be the leading coefficients of and repectively. Then the leading coefficient of is Thus Since is a unit, we’ll get which contradicts Therefore

*Existence of the representation *: We only need to prove the claim for The proof is by induction over It is clear for Suppose that the claim is true for any If then choose and So we may assume that Let Therefore, since is a unit, we will have which will give us Now apply the induction hypothesis to each term to finish the proof.