= Rational point
{title2=$X(K)$}
For an <algebraic variety> $X$ defined over a <field> $K$, a K-<rational point> is a morphism $\operatorname{Spec}K\to X$ over $K$. In affine coordinates it is a solution of the defining equations with every coordinate in $K$; in projective coordinates it is a nonzero coordinate vector over $K$, considered up to common nonzero scaling. The set is denoted $X(K)$. Over a <number field>, having points over every completion need not imply having a <rational point>; for genus-one coverings this failure is recorded by the <Tate–Shafarevich group>.
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