Supersimplicity and quadratic extensions
The paper deals with the algebraic characterization of supersimple fields. Pillay conjectured that such a field \(K\) is perfect, bounded and pseudo algebraically closed, and with Poizat proved both perfection and boundedness. Thus it remains to prove or disprove pseudo algebraic closedness, in other words that every absolutely irreducible plane curve defined over \(K\) has a \(K\)-rational point. The paper shows that if \(K\) is a supersimple field admitting exactly one extension of degree 2 (up to isomorphism), then every elliptic curve \(E\) defined over \(K\) has an \(s\)-generic \(K\)-rational point, i.e., a point \(P\) in \(E(K)\) such that the \(SU\)-rank of \(P\) over \(F\) equals the \(SU\)-rank of \(K\), where \(F\) is some small set of parameters over which \(E\) is defined.
scientific article; zbMATH DE number 5529379