ON RATIONAL POINTS OF CERTAIN AFFINE HYPERSURFACES

摘要

Let F be a field with char F not equal 2, let a(1), ... ,a(n) is an element of F*, and let f is an element of F[y] be a monic polynomial of degree 2m. Let further S be an affine hypersurface over F determined by the equation f(y) = Sigma(n)(i=1) a(i) x(i)(2). In the first part of the paper we prove a certain version of Springer's theorem. Namely, we show that if the form psi similar or equal to 1, -a(1), ... , -a(n) is anisotropic and S has an L-rational point for some odd-degree extension L/F, then S has an L-rational point for some odd-degree extension L/F with [L : F] = m, and the last inequality is strict in general.