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Non-PAC fields whose Henselian closures are separably closed. (English) Zbl 0991.12004
The notion of a PAC-field originated in [J. Ax, Ann. Math. (2) 88, 239-271 (1968; Zbl 0195.05701)]. A field $$K$$ is called PAC if every nonvoid absolutely irrreducible variety $$V$$ over $$K$$ has a $$K$$-rational point. Let $$F$$ be an extension of a field $$K.$$ An equivalence class of valuations of $$F$$ which are trivial over $$K$$ is called a prime divisor of $$F/K.$$ Denote the set of all prime divisors of $$F/K$$ by $${\mathbb P}(F/K).$$ For each $$p\in {\mathbb P}(F/K)$$ denote the Henselian closure of $$F$$ with respect to $$p$$ by $$F_p.$$ Tensoring central simple finite dimensional $$F$$-algebras with $$F_p$$ defines a homomorphism: $$\text{res}_{p}:\text{Br}(F)\rightarrow \text{Br}(F_{p}).$$ Consider the direct product of all these homomorphisms: $\text{res}:\text{Br}(F)\rightarrow\prod_{p\in {\mathbb P}(F/K)}\text{Br}(F_{p}).$ It is said that $$F$$ satisfies the Hasse principle for Brauer groups if res is injective. The main result of the paper is
Theorem D. Let $$K_0$$ be either a finite field or a global field. Then $$K_0$$ has an infinite regular extension $$K$$ such that (a) Every extension $$F$$ of $$K$$ of transcendence degree 1 satisfies the Hasse principle for Brauer groups. (b) $$K$$ is not formally real. (c) Each Henselization of $$K$$ is separably closed. (d) $$K$$ is not PAC.
The theorem gives solutions to two problems raised in [I. Efrat, Isr. J. Math. 122, 43-60 (2001; Zbl 1013.12003) and M. D. Fried and M. Jarden, Field Arithmetic, Ergebnisse der Mathematik (3) 11, Springer (1986; Zbl 0625.12001)].
Reviewer: G.Pestov (Tomsk)

##### MSC:
 12F20 Transcendental field extensions 12J05 Normed fields 16K50 Brauer groups (algebraic aspects) 14F22 Brauer groups of schemes
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