Generalizing a theorem of Richard Brauer

There exists a function image such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension greater-or-equal, slantedf(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.