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  "body": "\n## Definition\n\nFor a prime $p$, let $\\Q_p$ be the field of [[p-adic numbers|BT6T702C28]], and let $\\Q_\\infty = \\R$.\nLet $\\mathcal{C}$ be a class of systems of polynomial equations with coefficients in $\\Q$.\nFor a system of homogeneous equations, a solution means a solution other than the zero solution.\nLet $V$ be the set of all primes together with the symbol $\\infty$.\nThe class $\\mathcal{C}$ *satisfies the Hasse principle* if each system in $\\mathcal{C}$ with a solution in $\\Q_v$ for every $v \\in V$ has a solution in $\\Q$.\n\n## Remarks\n\nThe converse always holds, since $\\Q$ is a subfield of each $\\Q_v$.\nThe same definition makes sense over any number field, with its completions in place of the $\\Q_v$.\n\nExample: the class of homogeneous quadratic equations $Q(x_1, \\dots, x_n) = 0$, with $Q$ a quadratic form over $\\Q$, satisfies the Hasse principle; this is the [[Hasse–Minkowski theorem|R8HCWA981S]].\n\nNon-example: the class of homogeneous cubic equations in three variables does not.\nSelmer's equation $3x^3 + 4y^3 + 5z^3 = 0$ has nonzero solutions in $\\R$ and in every $\\Q_p$, but it has no nonzero solution in $\\Q$.\n",
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