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Let $(K, v)$ be a Henselian discrete valued field with a quasifinite residuefield. 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 Brauergroup Br$(E ^{\prime })$ is trivial, for every algebraic extension $E ^{\prime}/E$; (ii) finite extensions of $E$ are not $C _{1}$-fields. This, applied tothe maximal algebraic extension $K$ of the field $\mathbb{Q}$ of rationalnumbers in the field $\mathbb{Q} _{p}$ of $p$-adic numbers, for a given prime$p$, proves the existence of an algebraic extension $E _{p}/\mathbb{Q}$, suchthat dim$(E _{p}) \le 1$, $E _{p}$ is not a $C _{1}$-field, and $E _{p}$ has aHenselian valuation of residual characteristic $p$.

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