Wiesend, Götz Local-global principles for the Brauer group. (Lokal-Global-Prinzipien für die Brauergruppe.) (German) Zbl 0846.12007 Manuscr. Math. 86, No. 4, 455-466 (1995). 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. Reviewer: N.Schwartz (Passau) Cited in 3 Documents MSC: 12J10 Valued fields 12G05 Galois cohomology 12J12 Formally \(p\)-adic fields 14H05 Algebraic functions and function fields in algebraic geometry Keywords:local-global principle; Brauer group; real closed fields; \(p\)-adically closed fields; regular function fields PDFBibTeX XMLCite \textit{G. Wiesend}, Manuscr. Math. 86, No. 4, 455--466 (1995; Zbl 0846.12007) Full Text: DOI EuDML References: [1] N. Bourbaki, Algèbre, Chapitre VIII, Paris, 1958 [2] J.W.S. Cassels, A. Fröhlich, Algebraic number theory, Washington D.C., 1967 [3] O. Endler, Valuation Theory, Berlin 1972 · Zbl 0257.12111 [4] M. Fried, M. Jarden, Field Arithmetic, Berlin 1986 [5] Hartshorne, Algebraic Geometry. Springer, Berlin Heidelberg, 1977 [6] H. Hasse, R. Brauer und E. Noether, Beweis eines Hauptsatzes in der Theorie der Algebren. Journal für Mathematik, Band 167, 399–404 · JFM 58.0142.03 [7] U. Jensen, H. Lenzing, Model theoretic algebra, Amsterdam, 1989 [8] S. Lang, Abelian Varieties, New York, Interscience 1959 [9] S. Lang, On quasialgebraic closure, Annals of Mathematics55 (1952), 373–390 · Zbl 0046.26202 [10] S. Lichtenbaum, Duality Theorems for curves overp-adic fields, Invent. math.7 (1969), 120–136 · Zbl 0186.26402 [11] O.T. O’Meara, Introduction to quadratic forms. Springer, Berlin Göttingen Heidelberg, 1963 [12] F. Pop, Galoissche Kennzeichnungp-adisch abgeschlossener Körper, Journal reine angewandte Mathematik392 (1988), 145–175 · Zbl 0671.12005 [13] A. Prestel, P. Roquette, Formallyp-adic fields, Lecture Notes in Mathematics 1050, Berlin-Heidelberg-New York, 1984 · Zbl 0523.12016 [14] B.v. Querenburg, Mengentheoretische Topologie, Berlin, Heidelberg, New York, 1979 [15] P. Roquette, On the Galois Cohomology of the Projective Linear Group and its Applications to the Construction of Generic Splitting Fields of Algebras, Math. Annalen150 (1963), 411–439 · Zbl 0114.02206 [16] J.P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics5, Berlin-Heidelberg-New York 1973 [17] J.P. Serre, Local Fields, New York, 1979 · Zbl 0423.12016 [18] C. Tsen, Divisionsalgebren über Funktionenkörper, Nachr. Ges. Wiss. Göttingen (1933), 335 · JFM 59.0160.01 [19] E. Witt, Zerlegung reeller algebraischer Funktionen in Quadrate, Schiefkörper über reellen Funktionenkörpern, Journal reine angewandte Mathematik171 (1934), 4–11 · JFM 60.0099.01 [20] E. Witt, Über ein Gegenbeispiel zum Normensatz, Math. Zeitschrift39 (1935), 462–467 · Zbl 0010.14901 [21] O. Zariski, P. Samuel, Commutative Algebra, New Jersey, 1960 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.