On parametric and generic polynomials with one parameter

Given fields k ? L , our results concern one parameter L-parametric polynomials over and their relation to generic polynomials. The former are P ( T Y ) ? [ ] of group G which parametrize all Galois extensions via specialization in the latter those -parametric for every field ? . We show, example, that being with taken be single C V U is fact sufficient a polynomial generic. As corollary, we obtain complete list given characteristic 0, complementing classical literature on topic. Our approach also applies an old problem Schinzel: subject Birch Swinnerton-Dyer conjecture, provide families affine curves number fields, rational point, but no point.