Abelian lifts of polynomials

Abstract

Let p be a prime, and let f∈Z[x] be a non-constant polynomial whose leading coefficient is prime to p and that f has no repeated root modulo p. Suppose the mod-p irreducible factors of f all have the same degree. Then there are infinitely many Abelian extensions of number fields L/K such that OL/pOL≃(Z/p)[x]/f, where OL denotes the ring of integers of L and p⊂OK is a maximal ideal with OK/pOK≃Z/p. Moreover, under the generalized Riemann hypothesis for the Dedekind zeta functions of number fields, for fixed deg⁡(f) we can construct Abelian lifts in polynomial time in log⁡p. This confirms in a stronger form a question raised by Mihăilescu and Vuletescu concerning polynomial cyclic algebras. The proofs make use of a weak form of Artin's conjecture on primitive roots and polynomial time polynomial factoring algorithms over number fields.