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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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