Generic Polynomials with Few Parameters

Abstract

We call a polynomial g(t 1 , . . . , tm, X) over a field K generic for a group G if it has Galois group G as a polynomial in X, and if every Galois field extension N/L with K ⊆ L and Gal(N/L) ≤ G arises as the splitting field of a suitable specialization g(λ 1 , . . . , λm, X) with λ i ∈ L. We discuss how the rationality of the invariant field of a faithful linear representation leads to a generic polynomial which is often particularly simple and therefore useful. Then we consider various examples and applications in characteristic 0 and in positive characteristic. These include results on so-called vectorial polynomials and a generalization of an embedding criterion given by Abhyankar. We give recursive formulas for generic polynomials over a field of defining characteristic for the groups of upper unipotent and upper triangular matrices, and explicit formulae for generic polynomials for the groups GU 2 (q 2 ) and GO 3 (q).

Generic and Explicit Realization of Smallp-groups

“…We assume all fields to have characteristic = p. For the characteristic p case, we refer to Kemper and Mattig (2000), in this volume.…”

Generic extensions and generic polynomials for semidirect products

“…For SL(2, 3), Rikuna [9] proved that the invariant field of a four-dimensional representation is purely transcendental, so a generic polynomial can be constructed, e.g. using the methods of Kemper and Mattig [5].…”

Finite groups of essential dimension one

“…However, we will not make use of this connection in this paper. We refer to [10] or [8] for details.…”