Polynomial ring/Field/One variable/Definition

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

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

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

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