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

Abstract

Let $(K, v)$ be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension $E/K$ satisfying the following: (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) finite extensions of $E$ are not $C _{1}$-fields. This, applied to the maximal algebraic extension $K$ of the field $\mathbb{Q}$ of rational numbers in the field $\mathbb{Q} _{p}$ of $p$-adic numbers, for a given prime $p$, proves the existence of an algebraic extension $E _{p}/\mathbb{Q}$, such that dim$(E _{p}) \le 1$, $E _{p}$ is not a $C _{1}$-field, and $E _{p}$ has a Henselian valuation of residual characteristic $p$.