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.

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We consider (Frobenius) difference equations over (Fq(s, t), φq) where φq fixes t and acts on Fq(s) as the Frobenius endomorphism. We prove that every semisimple, simplyconnected linear algebraic group G defined over Fq can be realized as a difference Galois group over (F q i(s, t), φ q i ) for some i ∈ N. The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori's Theorem which states that G(Fq) occurs as (finite) Galois group over Fq(s).

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A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.