= Nonparametrizability of an elliptic curve
An <elliptic curve> in characteristic zero admits no nonconstant <rational map of projective varieties> from the <projective line>. Over $\mathbb C$, write its equation as $y^2=\prod_{j=1}^3(x-e_j)$ with distinct $e_j$. Substituting $x=u/v$ with <coprime polynomials> gives $(yv^2)^2=v\prod_j(u-e_jv)$. These four pairwise coprime factors must all be squares. The <polynomial pencil with four square members> forces $u,v$ constant, ruling out a nonconstant <rational parametrization of an algebraic curve>.
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