Polynomial ring/Field/n variables/Directly/Definition

For a field ${\displaystyle {}K}$ and a set of variables ${\displaystyle {}X_{1},\ldots ,X_{n}}$, the polynomial ring

${\displaystyle K[X_{1},\ldots ,X_{n}]}$

consists of all polynomials ${\displaystyle {}F(X_{1},\ldots ,X_{n})}$ in these variables. This set is made into a commutative ring by componentwise addition and by the multiplication that extends the rule

${\displaystyle {}X_{1}^{r_{1}}\cdots X_{n}^{r_{n}}\cdot X_{1}^{s_{1}}\cdots X_{n}^{s_{n}}:=X_{1}^{r_{1}+s_{1}}\cdots X_{n}^{r_{n}+s_{n}}\,}$

such that distributivity holds.