Generic polynomials and generic extensions in positive characteristic
Speaker: Professor Jorge Morales (LSU)
Time: 4.00PM
Date: Monday March 31st 2014
Location: CASL Seminar Room, Block 8, Belfield Office Park
Abstract:
Let G be a finite group, let k be a field and let t_1, ..., t_m be indeterminates over k. A separable polynomial f(X; t_1, ..., t_m) in k(t_1, ..., t_m)[X] is called generic for G over k if Gal(f/k(t_1, ..., t_m)) is isomorphic to G and if every Galois G-extension M/L (with k contained in L) is the splitting field of a suitable specialization of f at a point (a_1, ..., a_m) in L^m. Generic polynomials may fail to exist even for fairly elementary groups. For instance, there is no generic polynomial for the cyclic group C_8 over Q (Lenstra). We give explicit constructions of both generic polynomials and generic extensions (in the sense of Saltman) for certain families of finite groups of the form G=\mathbb{G}(F_q) where \mathbb{G} is a connected linear algebraic group defined over a finite field F_q and k is an infinite field containing F_q. These constructions use Matzat's theory of Frobenius modules. This is collaborative work with A. Sanchez and D. Tseng.
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