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

Quasifinite fields of prescribed characteristic and Diophantine dimension

“…The validity of the equality ddim(E) = ∞ in case (ii) can be deduced from its validity in case (i), by assuming the opposite and applying the lemma on page 379 of [15] (to suitably chosen forms without nontrivial zeroes over the completion of E with respect to some of its discrete valuations). Examples of algebraic extensions E 0 /Q and E p /Q p , p ∈ P, such that dim(E p ′ ) ≤ 1 < ddim(E p ′ ), for each p ′ ∈ P, can be found in [5]. At the same time, the question of whether there exist algebraic extensions E 0,ℓ /Q and E p,ℓ /Q p , p ∈ P, with ddim(E p ′ ,ℓ ) = ℓ, for each p ′ ∈ P, seems to be open, for any integer ℓ ≥ 2.…”

Diophantine problems over tamely ramified fields

“…85 [Ser79]). (iii) There are plenty of examples of algebraic extensions of local fields which are of cohomological dimension 1 but not C 1 (see [Chi22]). It would be interesting to investigate to what extent the maximal unramified extension and the maximal totally ramified extensions are the only minimal algebraic extensions of a local field which are C 1 , i.e., which do not contain any proper subfields which are C 1 .…”