Fields of dimension one algebraic over a global or local field need not be of type $(C_{1})$

Ivan D. Chipchakov

Published 2019-05-16Version 1

Let $(K, v)$ be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension $E/K$ with the following properties: (i) $E$ has dimension dim$(E) \le 1$, i.e. the Brauer group Br$(E ^{\prime })$ is trivial, for every algebraic extension $E ^{\prime }/E$; (ii) no finite extension of $E$ is a $C _{1}$-field, in the sense of E. Artin and Lang. This, applied to the case where $K$ is the maximal algebraic extension of the field $\mathbb Q$ of rational numbers in the field $\mathbb Q _{p}$ of $p$-adic numbers, for a given prime number $p$, proves the existence of an algebraic extension $E _{p}$ of $\mathbb Q$, such that dim$(E _{p}) \le 1$, $E _{p}$ has a Henselian valuation with a residue field of characteristic $p$, and $E _{p}$ is not a $C _{1}$-field.