Tsen's theorem

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. This implies that the Brauer group of any such field vanishes, and more generally that all the Galois cohomology groups "H" "i"("K", "K"*) vanish for "i" ≥ 1. This result is used to calculate the etale cohomology groups of an algebraic curve. The theorem was proved by Chiungtze C. Tsen (1933).

References

*Ding, Shisun; Kang, Ming-Chang; Tan, Eng-Tjioe "Chiungtze C. Tsen (1898-1940) and Tsen's theorems." Rocky Mountain J. Math. 29 (1999), no. 4, 1237-1269.
*Serge Lang, [http://links.jstor.org/sici?sici=0003-486X%28195203%292%3A55%3A2%3C373%3AOQAC%3E2.0.CO%3B2-W "On Quasi Algebraic Closure"] The Annals of Mathematics 2nd Ser., Vol. 55, No. 2 (Mar., 1952), pp. 373-390
*J.-P. Serre, "Galois cohomology", ISBN 3540421920
*C. Tsen, "Divisionsalgebren über Funkionenkörper", Nachr. Ge. Wiss. Gottingen (1933) p. 335


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