On the anisotropic splitting of division algebrasSpeaker:Professor Ulf Rehmann (Universitaet Bielefeld)
Date: Thu 19th April 2012
Location: Mathematical Sciences Seminar Room (Ag 1.01)
To understand division algebras, a classical method is to simplify their structure by extending their base field, e.g., algebras can be "split", that is, made isomorphic to a full matrix ring over a suitable extension.
It is more interesting to find field extensions for which a given division algebra stays "anisotropic", i.e., it remains a division algebra over that extension, but of possibly simpler structure.
There are two interesting recent results:
1. A theorem of Hasse-Brauer-Noether states that every central simple algebra over a number field is cyclic. This does not hold for arbitrary fields. However, we have the following result: For any given field F there exists a regular field extension E/F such that i) any central simple E-algebra is cyclic, ii) for any central simple F-algebra, index and exponent over E (after field extension) are the same as over F, iii) the restriction homomorphism res Br(F) --> Br(E) is injective.
2. For given "disjoint" algebras A_1, ... A_n and any set of "admissible" values for indices and exponents for every A_i one can construct a (finitely generated) regular field extension of the ground field so that index and exponent of A_i over that extension will attain the prescribed values.
This will be discussed in the talk. (Results based on joint work with S. Tikhonov and V. Yanchevskii.)
(This talk is part of the K-Theory, Quadratic Forms and Number Theory series.)