Fields of dimension one algebraic over a global or local field need not be of type C1

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 C-1-fields. This, applied to the maximal algebraic extension K of the field Q of rational numbers in the field Q(p) of p-adic numbers, for a given prime p, proves the existence of an algebraic extension E-p/Q, such that dim(E-p) <= 1, E-p is not a C-1-field, and E-p has a Henselian valuation of residual characteristic p. (C) 2021 Elsevier Inc. All rights reserved.