  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 $p$.
