Polynomial ring/Field/One variable/Definition

Polynomial ring

The polynomial ring over a field ${}K$ consists of all polynomials

${}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,,$

where ${}a_{i}\in K$, ${}n\in \mathbb {N}$. It is endowed with componentwise addition and a multiplication, which arises by distributive continuation of the rule

${}X^{n}\cdot X^{m}:=X^{n+m}\,.$