Teichmüller cocycle

In mathematics, the Teichmüller cocycle is a certain 3-cocycle associated to a simple algebra A over a field L which is a finite Galois extension of a field K and which has the property that any automorphism of L over K extends to an automorphism of A. The Teichmüller cocycle, or rather its cohomology class, is the obstruction to the algebra A coming from a simple algebra over K. It was introduced by Teichmüller (1940) and named by Eilenberg and MacLane (1948).

Properties

If K is a finite normal extension of the global field k, then the Galois cohomology group H³(Gal(K/k,K*) is cyclic and generated by the Teichmüller cocycle. Its order is n/m where n is the degree of the extension K/k and m is the least common multiple of all the local degrees (Artin Tate).

References

• Artin, Emil; Tate, John (2009) [1952], Class field theory, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4426-7, https://books.google.com/books?isbn=978-0-8218-4426-7 
• Eilenberg, Samuel; MacLane, Saunders (1948), "Cohomology and Galois theory. I. Normality of algebras and Teichmüller's cocycle.", Trans. Amer. Math. Soc. 64: 1–20, doi:10.1090/s0002-9947-1948-0025443-3 
• Teichmüller, Oswald (1940), "Über die sogenannte nichtkommutative Galoissche Theorie und die Relation [math]\displaystyle{ \xi_{\lambda,\mu,\nu}\xi_{\lambda,\mu\nu,\pi}\xi^{\lambda}_{\mu,\nu,\pi}=\xi_{\lambda,\mu,\nu \pi}\xi_{\lambda \mu,\nu,\pi} }[/math]", Deutsche Mathematik: 138–149