In mathematics, Tsen's theorem states that a function field ''K'' of an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

over an algebraically closed field is

quasi-algebraically closed In mathematics, a field ''F'' is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebra ...

(i.e., C₁). This implies that the

Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...

of any such field vanishes, and more generally that all the

Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natur ...

groups ''H''^''i''(''K'', ''K''^*) vanish for ''i'' ≥ 1. This result is used to calculate the

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectur ...

groups of an algebraic curve. The theorem was published by

Chiungtze C. Tsen Chiungtze C. Tsen (; Chang-Du Gan: sɛn˦˨ tɕjuŋ˨˩˧ tsɹ̩˦˨ April 2, 1898 – October 1, 1940), given name Chiung (), was a Chinese mathematician born in Nanchang, Jiangxi. He is known for his work in algebra. He was one of Emmy No ...

in 1933.

References

* * * * Theorems in algebraic geometry {{algebraic-geometry-stub

See also

References