- Properties
- References
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 {{harvs|txt|authorlink=Oswald Teichmüller|last=Teichmüller |year=1940}} and named by {{harvs|txt|last1=Eilenberg|last2=MacLane|year=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 {{harv|Artin|Tate|2009|loc=p.68}}.
References
- {{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | last2=Tate | first2=John | author2-link=John Tate | title=Class field theory | origyear=1952 | url=https://books.google.com/books?isbn=978-0-8218-4426-7 | publisher=AMS Chelsea Publishing, Providence, RI | isbn=978-0-8218-4426-7 | mr=0223335 | year=2009}}
- {{citation|mr=0025443
|last=Eilenberg|first= Samuel|last2= MacLane|first2= Saunders
|title=Cohomology and Galois theory. I. Normality of algebras and Teichmüller's cocycle.
|journal=Trans. Amer. Math. Soc.|volume= 64|year=1948|pages= 1–20|doi=10.1090/s0002-9947-1948-0025443-3}}
- {{citation|last=Teichmüller|first= Oswald|title=Über die sogenannte nichtkommutative Galoissche Theorie und die Relation
|journal=Deutsche Mathematik|year=1940|pages=138–149}}