Diego Izquierdo

On a conjecture of Kato and Kuzumaki

In 1986, Kato and Kuzumaki stated several conjectures in order to give a diophantine characterization of cohomological dimension of fields. In this article, we first prove a local-global principle in this context for number fields. This allows us to give a new proof of one of Kato and Kuzumaki’s conjectures for totally imaginary number fields (the first proof was given by Olivier Wittenberg). Our arguments can be generalized to get results for global fields of positive characteristic. We then establish all the conjectures for the fields C(x1, ..., xn) and C(x1, ..., xn)((t)). We finally prove a partial result for the field of Laurent series in two variables C((x, y)).