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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.

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