Polynomial Ring

algebra

Definition

Polynomial Ring

Let $F$ be a field (or more generally a commutative ring). The polynomial ring $F[x]$ in one indeterminate $x$ over $F$ is the set of all polynomials with coefficients in $F$, equipped with pointwise addition and Cauchy-product multiplication.

Formally, $F[x]$ consists of all sequences $(a_0,a_1,a_2,\dots )$ in $F$ with finite support, where

$$\begin{array}{rl}(a_i)+(b_i)& =(a_i+b_i),\\ (a_i)\cdot (b_i)& =\left(\sum _{j=0}^i{a}_j{b}_{i-j}\right).\end{array}$$

With these operations, $F[x]$ is a commutative ring with unity $1_F$.