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  "body": "\n## Statement\n\nLet $Q$ be a nondegenerate [[quadratic form|0RBQS1JJCR]] in $n$ variables with coefficients in $\\Q$.\nFor a prime $p$, let $\\Q_p$ be the field of [[p-adic numbers|BT6T702C28]].\nThen $Q$ has a zero $x \\in \\Q^n$ with $x \\ne 0$ if and only if both of these hold:\n\n1. $Q$ has a nonzero zero in $\\R^n$.\n2. $Q$ has a nonzero zero in $\\Q_p^n$ for every prime $p$.\n\n## Remarks\n\nThe direction \"only if\" is clear; the content is that local zeros give a global zero.\nThe proof goes by the number of variables, with the [[Hilbert symbol|WFG4YMBYVQ]] as the main tool.\nThe case $n = 3$ is essentially Legendre's theorem on $ax^2 + by^2 + cz^2 = 0$; the case $n = 4$ uses [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]], and the case $n \\ge 5$ reduces to $n = 4$.\nThe theorem holds over every number field, with its completions in place of $\\R$ and the $\\Q_p$.\nExample: $x^2 + y^2 - 3z^2$ has the real zero $(\\sqrt{3}, 0, 1)$, but no nonzero zero in $\\Q_3^3$; hence it has no nonzero rational zero.\nA consequence: a nonzero $a \\in \\Q$ is represented by $Q$ over $\\Q$ if and only if it is represented over $\\R$ and over every $\\Q_p$.\nAnother: two quadratic forms over $\\Q$ are equivalent over $\\Q$ if and only if they are equivalent over $\\R$ and over every $\\Q_p$.\nThe restriction to quadratic forms is necessary: Selmer's cubic $3x^3 + 4y^3 + 5z^3$ has nonzero zeros over $\\R$ and every $\\Q_p$, but none over $\\Q$.\n\n## Explanations\n\n### L1 human\n\nTo decide whether a quadratic equation such as $ax^2 + by^2 + cz^2 = 0$ has a solution in fractions, it is enough to check real numbers and the $p$-adic numbers for each prime.\nEach such check is a finite computation, and only finitely many primes need real work.\nSo the theorem turns an infinite search into a finite test.\n\n### L2 human\n\nThis is the main example of the [[Hasse principle|EMAR4X3W8D]].\nIn practice one computes Hilbert symbols $(a, b)_p$ at the primes dividing $2$ and the coefficients; at all other primes a form of rank at least $3$ is automatically isotropic.\nThe theorem is the basis of the classification of quadratic forms over $\\Q$ by discriminant, signature and Hasse invariants.\nFor cubic and higher degree equations the principle can fail, and the study of the failure leads to the Tate–Shafarevich group of elliptic curves.\n",
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    "name": "Hasse–Minkowski theorem",
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  "msc": [
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  "name": "Hasse–Minkowski theorem",
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