= Local solubility
= Locally soluble
{synonym}
A <Diophantine equation> is soluble at a place of a <number field> if it has a solution over the corresponding <completion of a valued field>. It is locally soluble everywhere if this holds at all finite and infinite places. Over the <rational numbers>, these are the real field and all <p-adic fields>. Local solubility is necessary for a <rational point>, but the <local-to-global gap in isogeny descent> shows it need not suffice.
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