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Tsen's theorem
In mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes,[1] and more generally that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1. This result is used to calculate the étale cohomology groups of an algebraic curve.
The theorem was proved by Zeng Jiongzhi (also rendered as Chiungtze C. Tsen in English) in 1933.
See also
Tsen rank
References
Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. p. 181. ISBN 978-0-387-72487-4. Zbl 1130.12001.
Ding, Shisun; Kang, Ming-Chang; Tan, Eng-Tjioe (1999), "Chiungtze C. Tsen (1898–1940) and Tsen's theorems", Rocky Mountain Journal of Mathematics 29 (4): 1237–1269, doi:10.1216/rmjm/1181070405, ISSN 0035-7596, MR 1743370, Zbl 0955.01031
Lang, Serge (1952), "On quasi algebraic closure", Annals of Mathematics (2nd Ser.) 55: 373–390, ISSN 0003-486X, Zbl 0046.26202
Serre, J. P. (2002), Galois Cohomology, Springer Monographs in Mathematics, Translated from the French by Patrick Ion, Berlin: Springer-Verlag, ISBN 3-540-42192-0, Zbl 1004.12003
Tsen, Chiungtze C. (1933), "Divisionsalgebren über Funktionenkörpern", Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. (in German): 335–339, JFM 59.0160.01, Zbl 0007.29401
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