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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 \( f(x), g(x) \in R[x] \) be polynomials, with \( f(x) \) being a monic polynomial.

Then there exists a unique pair of polynomials \( q(x), r(x) \in R[x] \) where \( \operatorname{deg} r < \operatorname{deg} f \) 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 \( f(x), g(x) \in R[x] \) be polynomials, with \( f(x) \) being a monic polynomial.

Then there exists a unique polynomial \( r(x) \in R[x] \) where \( \operatorname{deg} r < \operatorname{deg} f \) 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?