Polynomial ring

In algebra, the polynomial ring over a commutative ring is a ring which formalises the polynomials of elementary algebra.

Construction of the polynomial ring
Let R be a ring. Consider the R-module of sequences


 * $$\left(a_0, a_1, \ldots, a_n, \ldots \right) \,$$

which have only finitely many non-zero terms, under pointwise addition


 * $$(a+b)_n = a_n + b_n .\,$$

We define the degree of a non-zero sequence (an) as the the largest integer d such that ad is non-zero.

We define "convolution" of sequences by


 * $$(a \star b)_n = \sum_{i=0}^n a_i b_{n-i} .\,$$

Convolution is a commutative, associative operation on sequences which is distributive over addition.

Let X denote the sequence


 * $$X = (0,1,0,\ldots) .\,$$

We have


 * $$X^2 = (0,0,1,0,\ldots) \,$$
 * $$X^3 = (0,0,0,1,0,\ldots) \,$$

and so on, so that


 * $$ \left(a_0, a_1, \ldots, a_n, \ldots \right) = \sum_n a_n X^n ,\,$$

which makes sense as a finite sum since only finitely many of the an are non-zero.

The ring defined in this way is denoted $$R[X]$$.

Properties

 * If R is commutative then so is R[X].
 * If R is an integral domain then so is R[X].
 * In this case the degree function satisfies $$\deg(fg) = \deg(f) + \deg(g)$$.
 * If R is a unique factorisation domain then so is R[X].
 * If R is a Noetherian domain then so is R[X].
 * If R is a field, then R[X] is a Euclidean domain.