Generic Polynomials in Few Parameters for Non-Abelian Groups of Degree 8 and 9

The notion of generic polynomial was introduced by DeMeyer for a group G to be a multivariable polynomial that gives all G-extensions over a field k as splitting fields of specializations of the variables. Thus a generic polynomial is nearly equivalent to having a generic G-extension with the added concreteness of a polynomial. In general a generic polynomial need not exist, failing most notably for abelian groups of order divisible by eight over a field of characteristic zero. General theory has given the existence in many cases and bounds on the number of parameters needed. The minimum number of parameters is an invariant of general interest called the generic dimension.

Our goal is to find explicitly generic polynomials for certain non-abelian groups of degree eight and nine, namely G₁ the cyclic group of order two semidirect product the elementary abelian group of order nine, G₂ the modular group of order sixteen, and G₃=SL(2,3). We follow the method of Kemper and Mattig to compute generic polynomials in arbitrary fields of characteristic relatively prime to the groups' orders. From our computations we establish that the generic dimension of G₁ is two. The existence of a generic polynomial for G₂ in five parameters is known by Ledet; we lower this number to four. Finally, in an unpublished work from 2000 by Rikuna, the rationality of a four dimensional representation is established for G₃. We seek to extend a computation of Groebner for Q₈ to a different construction of a generic polynomial for G₃.