, Volume 59, Issue 3, pp 842-849

Factorization of solvable polynomials over finite fields and the generalized Riemann hypothesis

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Abstract

This article presents an algorithm that, assuming the generalized Riemann hypothesis, factors a polynomial f mod p, where f ∃Z[X] is solvable overQ, into irreducible (over the fieldF pm) factors in time polynomial in m, log p, and the length of notation of f. The following problems are also solved in time polynomial in m, n, and log p: 1) construction of the fieldF pm, 2) construction of all isomorphisms between two realizations ofF p m, and 3) computation of the roots of degree n inF p m.

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Mathematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 176, pp. 104–117, 1989.