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Local-global principles for the Brauer group. (Lokal-Global-Prinzipien für die Brauergruppe.) (German) Zbl 0846.12007
Suppose that \(K\) is a field and that \(I\) is some set of local objects like valuations or places. For each \(i\in I\) let \(K_i\) be some extension field of \(K\) which may be a Henselization or a completion. One says that \(K\) has a local-global principle for the Brauer group with respect to \(I\) if the canonical map \(\text{Br} (K)\to \prod \text{Br} (K_i)\) is injective. For global fields the Hasse-Brauer-Noether theorem is a classical result of this type. More recently, F. Pop obtained results about function fields in one variable over real closed fields or \(p\)-adically closed fields. The author shows by examples that there are fields having no local-global principle. He studies an elementary class of fields with the property that all regular function fields in one variable have a local-global principle. This class of fields includes real closed and \(p\)-adically closed fields.

12J10 Valued fields
12G05 Galois cohomology
12J12 Formally \(p\)-adic fields
14H05 Algebraic functions and function fields in algebraic geometry
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