Fields of cohomological dimension 1 versus C₁-fields

Abstract

Ax gave examples of fields of cohomological dimension 1 which are not C₁-fields. Kato and Kuzumaki asked whether a weak form of the C₁-property holds for all fields of cohomological dimension 1 (existence of solutions in extensions of coprime degree rather than existence of a solution in the ground field). Using work of Merkur’ev and Suslin, and of Rost, D. Madore and I recently produced examples which show that the answer is in the negative. In the present note, I produce examples which require less work than the original ones. In the original paper, some of the examples were given by forms of degree 3 in 4 variables. Here, for an arbitrary prime p ≥ 5, I use forms of degree p in p + 1 variables.