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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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