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)≤1, i.e. the Brauer group Br(E′) is trivial, for every algebraic extension E′/E; (ii) finite extensions of E are not C1-fields. This, applied to the maximal algebraic extension K of the field Q of rational numbers in the field Qp of p-adic numbers, for a given prime p, proves the existence of an algebraic extension Ep/Q, such that dim(Ep)≤1, Ep is not a C1-field, and Ep has a Henselian valuation of residual characteristic p.