Polynomial ring

Definition

Let $R$ be a ring. A polynomial over $R$ is a sequence $(a_0, a_1, a_2, \dots)$ of elements of $R$ such that $a_n = 0$ for all but finitely many $n$. The element $a_n$ is the coefficient of degree $n$. The polynomial ring $R[x]$ is the set of these polynomials, with these operations: $$(a_n)_n + (b_n)_n = (a_n + b_n)_n, \qquad (a_n)_n \cdot (b_n)_n = (c_n)_n, \quad c_n = \sum_{i + j = n} a_i b_j.$$ With these operations, $R[x]$ is a ring.

Remarks

Let $x$ be the polynomial $(0, 1, 0, 0, \dots)$, and identify $r \in R$ with $(r, 0, 0, \dots)$. Then each polynomial with $a_n = 0$ for $n > d$ is $a_0 + a_1 x + \dots + a_d x^d$, in exactly one way. If $R$ is a commutative ring, then $R[x]$ is commutative. A polynomial is a formal expression and not a function: over $\mathbb{F}_2$, $x^2 + x$ is a nonzero polynomial, but it is zero at both elements of $\mathbb{F}_2$.

Explanations

L1 human

A polynomial is a finite list of coefficients. You add polynomials coefficient by coefficient. You multiply them with the rule $x^i \cdot x^j = x^{i + j}$ and the distributive law.