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 Hⁱ(K, K^*) vanish for i ≥ 1. This result is used to calculate the étale cohomology groups of an algebraic curve.

The theorem was published by Chiungtze C. Tsen in 1933.

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, Second Series, 55 (2): 373–390, doi:10.2307/1969785, ISSN 0003-486X, JSTOR 1969785, 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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