Fields of dimension one algebraic over a global or local field need not be of type $C_1$

Let $(K,v)$ be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension $E\mathrm{/}K$ satisfying the following: (i) $E$ has dimension dim$(E)\le 1$, i.e. the Brauer group Br$(E^{\mathrm{\prime }})$ is trivial, for every algebraic extension $E ^{\prime }/E$;(ii)finiteextensionsof$E$arenot$C _{1}$-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 $E_p\mathrm{/}\mathbb{Q}$, 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 $p$.