Quasi-algebraically closed field
In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper ; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper . The idea itself is attributed to Lang's advisor Emil Artin. Formally, if P is a non-constant homogeneous polynomial in variables X1, ..., XN, and of degree d satisfying d < N then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have P(x1, ..., xN) = 0. In geometric language, the hypersurface defined by P, in projective space of degree , then has a point over F. Examples Any algebraically closed field is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero. Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem. Algebraic function fields of dimension 1 over algebraically closed fields are quasi-algebraically closed by