LAUSR

Abstract Let K be the algebraic closure of a finite field F q of odd characteristic p . For a positive integer m prime to p , let F = K ( x , y ) be the transcendence degree 1 function field defined by y q + y = x m + x − m . Let t = x m ( q − 1 ) and H = K ( t ) . The extension F | H is a non-Galois extension. Let K be the Galois closure of F with respect to H . By Stichtenoth [20] , K has genus g ( K ) = ( q m − 1 ) ( q − 1 ) , p -rank (Hasse–Witt invariant) γ ( K ) = ( q − 1 ) 2 and a K -automorphism group of order at least 2 q 2 m ( q − 1 ) . In this paper we prove that this subgroup is the full K -automorphism group of K ; more precisely Aut K ( K ) = Δ ⋊ D where Δ is an elementary abelian p -group of order q 2 and D has an index 2 cyclic subgroup of order m ( q − 1 ) . In particular, m | Aut K ( K ) | > g ( K ) 3 / 2 , and if K is ordinary (i.e. g ( K ) = γ ( K ) ) then | Aut K ( K ) | > g 3 / 2 . On the other hand, if G is a solvable subgroup of the K -automorphism group of an ordinary, transcendence degree 1 function field L of genus g ( L ) ≥ 2 defined over K , then | Aut K ( K ) | ≤ 34 ( g ( L ) + 1 ) 3 / 2 68 2 g ( L ) 3 / 2 ; see [15] . This shows that K hits this bound up to the constant 68 2 . Since Aut K ( K ) has several subgroups, the fixed subfield F N of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in Aut K ( K ) is large enough. This possibility is worked out for subgroups of Δ.