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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nA 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 $n$.\nThe element $a_n$ is the coefficient of degree $n$.\nThe polynomial ring $R[x]$ is the set of these polynomials, with these operations:\n$$(a_n)_n + (b_n)_n = (a_n + b_n)_n, \\qquad (a_n)_n \\cdot (b_n)_n = (c_n)_n, \\quad c_n = \\sum_{i + j = n} a_i b_j.$$\nWith these operations, $R[x]$ is a ring.\n\n## Remarks\n\nLet $x$ be the polynomial $(0, 1, 0, 0, \\dots)$, and identify $r \\in R$ with $(r, 0, 0, \\dots)$.\nThen each polynomial with $a_n = 0$ for $n \u003e d$ is $a_0 + a_1 x + \\dots + a_d x^d$, in exactly one way.\nIf $R$ is a [[commutative ring|0GYGHKX834]], then $R[x]$ is commutative.\nA 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 $\\mathbb{F}_2$.\n\n## Explanations\n\n### L1 human\n\nA polynomial is a finite list of coefficients.\nYou add polynomials coefficient by coefficient.\nYou multiply them with the rule $x^i \\cdot x^j = x^{i + j}$ and the distributive law.\n",
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