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 2009, p.68).

References

• Artin, Emil; Tate, John (2009) [1952], Class field theory, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4426-7, MR 0223335
• 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, MR 0025443
• Teichmüller, Oswald (1940), "Über die sogenannte nichtkommutative Galoissche Theorie und die Relation $\xi _{\lambda ,\mu ,\nu }\xi _{\lambda ,\mu \nu ,\pi }\xi _{\mu ,\nu ,\pi }^{\lambda }=\xi _{\lambda ,\mu ,\nu \pi }\xi _{\lambda \mu ,\nu ,\pi }$", Deutsche Mathematik: 138–149