Exceptional polynomials with 2-transitive affine monodromy groups
Let \(k\) denote a finite field with algebraic closure \(K\). A polynomial \(f(X)\in k[X]\) is called exceptional if \(K\) contains infinitely many finite extensions of \(k\), each of whose elements are permuted by \(f\). Let \(t\) denote a transcendental over \(k\). The main result of the paper under review is the classification of exceptional polynomials \(f(X)\) such that the polynomial \(f(X)-t\in k(t)[X]\) is separable and its Galois group \(A\) (the affine arithmetic monodromy group) acts doubly transitively on its roots. It is shown that these polynomials \(f\) belong to well-known families of exceptional polynomials. In particular, in the Galois group \(G\) of \(f(X)-t\) over \(K(t)\) (the affine geometric monodromy group) the stabilizer of any root is cyclic or dihedral. Such polynomials \(f\) are constructed explicitly in the paper.
scientific article; zbMATH DE number 6017555
scientific article