Additive polynomials for finite groups of Lie type

Abstract

This paper provides a realization of all classical finite groups of Lie type as well as a number of exceptional ones (with low-dimensional representations) as Galois groups over function fields over Fq and derives explicit additive polynomials for the extensions. Our unified approach is based on results of Matzat which give bounds for Galois groups of Frobenius modules and uses the structure and representation theory of the corresponding connected linear algebraic groups.

Generic extensions and generic polynomials for linear algebraic groups

“…is a set of strong generators (for the definition of strong generators, see [1] Section 3.4) (b) if B ∈ GL n (L), and (L n , Φ B ) has Galois group G(F q ), then B is Frobenius equivalent to A(ξ), for some ξ ∈ L d then G(F q ) admits a generic polynomial in d parameters over F q .…”

“…We use the special case of the Lower Bound Theorem as stated in the Albert-Maier paper [2,Theorem 3.4].…”

Generic extensions and generic polynomials for multiplicative groups

“…Most of the material in Sections 2.1-2.3 can be found in [9, Part I], [2]. We include it here for the convenience of the reader.…”

“…Let v = (1, 0, 1) T ∈ F 3 2 . The matrix N = v|Av (2) |AA (2) v (4) is nonsingular, so v is indeed a generator. As before, we compute Δ = N −1 AN (2) .…”

“…The matrix N = v|Av (2) |AA (2) v (4) is nonsingular, so v is indeed a generator. As before, we compute Δ = N −1 AN (2) . Recall that the entries in the last column of Δ are the coefficients of an additive generic polynomial f (Y ; t) of degree 8 for A × A 4 by Theorem 4.6.…”