Generalizing a theorem of Richard Brauer

Abstract

There exists a function f : N → N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension ⩾ f ( d ) , the set X ( K ) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois groups.