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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer.\nAn *integral quadratic form* in $n$ variables is a [[quadratic form|0RBQS1JJCR]]\n$$Q(x_1, \\dots, x_n) = \\sum_{1 \\le i \\le j \\le n} a_{ij} x_i x_j$$\nwhose coefficients $a_{ij}$ all lie in the [[integers|0C4JS9RF5M]].\nTwo integral quadratic forms $Q$ and $Q'$ in $n$ variables are *equivalent* if $Q'(x) = Q(Mx)$ for some $n \\times n$ integer [[matrix|0VD9ZSWWJG]] $M$ with $\\det M = \\pm 1$.\n\n## Remarks\n\nEvery integral quadratic form takes integer values at integer points.\nThe form can be written $Q(x) = \\frac12 x^T A x$ with a symmetric integer matrix $A$ whose diagonal entries $2a_{ii}$ are even.\nSome authors call $Q$ integral only when $Q(x) = x^T B x$ for a symmetric integer matrix $B$.\nThis means that all cross coefficients $a_{ij}$ with $i \u003c j$ are even; such forms are called classically integral.\n\nExample: $x^2 + xy + y^2$ is integral, with $a_{11} = a_{12} = a_{22} = 1$; it is not classically integral, because the coefficient of $xy$ is odd.\n\nNon-example: $x^2 + \\frac12 xy + y^2$ is a quadratic form over $\\Q$ but not integral, since the coefficient of $xy$ is not an integer; its value at $(1, 1)$ is $5/2$.\n",
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