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

Let $K$ be a global or local field, $E/K$ Galois extension, and Br$(E)$ the Brauer group of $E$. This paper shows that if is $v$ its natural discrete valuation, $v'$ valuation $E$ extending $v$, $q$ characteristic residue field $\widehat E$ $(E, v')$, then Br$(E) = \{0\}$ only following conditions hold: contains as subfield maximal $p$-extension K$, for each prime $p \neq q$; an algebraically closed in case value $v'(E)$ $q$-indivisible. When it characterizes fields with \{0\}$, which lie class tame abelian extensions $K$. We also give criterion that, latter case, any integer $n \ge 2$, there exists $n$-variate $E$-form degree $n$, violates Hasse principle.