Polynomials in several variables

Definition

Let $R$ be a commutative ring and let $n \ge 1$ be an integer. For $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n$, the monomial $x^\alpha$ is the formal product $x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. A polynomial in $x_1, \dots, x_n$ over $R$ is a finite sum $\sum_\alpha c_\alpha x^\alpha$ with $c_\alpha \in R$. Two polynomials are equal if and only if they have the same coefficient $c_\alpha$ for each $\alpha$. The ring $R[x_1, \dots, x_n]$ is the set of these polynomials, with coefficientwise addition and the product $$\Big(\sum_\alpha c_\alpha x^\alpha\Big) \Big(\sum_\beta d_\beta x^\beta\Big) = \sum_\gamma \Big(\sum_{\alpha + \beta = \gamma} c_\alpha d_\beta\Big) x^\gamma.$$

Remarks

There is a natural isomorphism $R[x_1, \dots, x_n] \cong R[x_1, \dots, x_{n-1}][x_n]$. So the ring can also be built by repeated use of the polynomial ring in one variable. The same construction works for any set of variables, with exponent maps that are zero at all but finitely many variables. The polynomial $x_1 x_2 + x_3^2$ is an element of $\mathbb{Z}[x_1, x_2, x_3]$.

Explanations

L1 human

A polynomial in several variables is a finite sum of terms like $5 x^2 y z^3$. You add and multiply such polynomials with the usual rules of algebra. Each term is a coefficient times a product of powers of the variables.