EXPLICIT REALIZATION OF THE DICKSON GROUPS G2(q) AS GALOIS GROUPS

In this paper we are concerned with the construction of polynomials whose Galois groups are the exceptional simple Chevalley groups G2(q), q a prime power, first discovered by Dickson; see Theorems 4.1 and 4.3. It was shown by Nori [7] that all semisimple simply-connected linear algebraic groups over Fq do occur as Galois groups of regular extension of regular function fields over Fq, but his proof does not give an explicit equation or even a constructive method for obtaining such extensions. On the other hand, in a long series of papers Abhyankar has given families of polynomials for groups of classical types (see [1] and the references cited there). His ad hoc approach hasn’t yet led to families with groups of exceptional type (but see [2] for a different construction of polynomials with Galois group the simple groups of Suzuki). Thus it seems natural to try to fill this gap. In his recent paper Matzat [6] describes an algorithmic approach which reduces the construction of generating polynomials for such extensions to certain group theoretic calculations. More precisely, let F := Fq(t), with t = (t1, . . . , ts) a set of indeterminates. We denote by φq : F → F , x → xq, the Frobenius endomorphism. Let G be a reduced connected linear algebraic group defined over Fq, with a faithful linear representation Γ : G(F ) ↪→ GLn(F ) in its defining characteristic, also defined over Fq. We identify G(F ) with its image in GLn(F ). Fix an element g ∈ G(F ) and assume that g ∈ GLn(R), where R := Fq[t]. Any specialization homomorphism ψ : R → Fqa , tj → ψ(tj), can be naturally extended to GLn(R). We define gψ := ψ(g) · ψ(φq(g)) · · ·ψ(φa−1 q (g)) ∈ GLn(Fq). With these notations Matzat [6, Thm. 4.3 and 4.5] shows the following: Theorem 1.1 (Matzat). Let G(F ) ≤ GLn(F ) be a reduced connected linear algebraic group defined over Fq. Let g ∈ GLn(R) such that: