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  "body": "\n## Summary\n\nThe Hasse principle, or local-global principle, concerns polynomial equations with rational coefficients.\nIt says that such an equation should have a rational solution when it has a real solution and a [[p-adic|BT6T702C28]] solution for each prime $p$.\nLocal solvability is easy to test, so when the principle holds it turns a hard question into finitely many checks.\nThe [[Hasse–Minkowski theorem|R8HCWA981S]] shows that it holds for quadratic forms.\nIt fails for some cubic equations, and the failures are measured by objects such as the [[Tate–Shafarevich group|BK9WBMF56G]].\n\n## Explanations\n\n### L1 human\n\nA solution in rational numbers gives a solution modulo every power of every prime, and a real solution.\nThese \"local\" solutions are necessary conditions, and they can be checked by finite computations.\nThe Hasse principle asks whether the necessary conditions are also enough.\nFor $x^2 + y^2 = 3z^2$ there is no solution modulo $9$ with $x$, $y$, $z$ not all divisible by $3$, so there is no nonzero rational solution either.\nFor quadratic equations, the local conditions are always enough; for the cubic $3x^3 + 4y^3 + 5z^3 = 0$ they are not.\n",
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