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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $n \\ge 1$ be an integer.\nLet $k\\langle x_1, \\dots, x_n, y_1, \\dots, y_n \\rangle$ be the [[free algebra|2P3Y2FRCSZ]] on $2n$ generators.\nLet $I$ be the [[two-sided ideal|0CD87SFM2N]] generated by the elements\n$$x_i x_j - x_j x_i, \\qquad y_i y_j - y_j y_i, \\qquad y_i x_j - x_j y_i - \\delta_{ij}$$\nfor $1 \\le i, j \\le n$, where $\\delta_{ij}$ is $1$ if $i = j$ and $0$ otherwise.\nThe *$n$-th Weyl algebra* is the [[quotient ring|0076HWHGHQ]] $A_n(k) = k\\langle x_1, \\dots, x_n, y_1, \\dots, y_n \\rangle / I$.\n\n## Remarks\n\nThe monomials $x_1^{a_1} \\cdots x_n^{a_n} y_1^{b_1} \\cdots y_n^{b_n}$ with $a_i, b_i \\ge 0$ form a basis of $A_n(k)$ over $k$.\nExample: in $A_1(k)$, with $y x = x y + 1$, one gets $y x^2 = (x y + 1) x = x (y x) + x = x(xy + 1) + x = x^2 y + 2x$.\nThis matches the rule $\\frac{d}{dx}(x^2 f) = x^2 f' + 2 x f$ for the operators of multiplication by $x$ and differentiation.\nNon-example: the commutative polynomial ring $k[x, y]$ satisfies the first two kinds of relations but has $y x - x y = 0 \\ne 1$.\nIf $k$ has characteristic $0$, then $A_n(k)$ is a simple Noetherian domain with center $k$.\nIf $k$ has characteristic $p \u003e 0$, then $x_1^p$ lies in the center of $A_n(k)$, and $A_n(k)$ is not simple.\nEquivalently, $A_1(k)$ is the Ore extension $k[x][y; \\id, d/dx]$.\n",
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