Factorization of polynomials over finite fields and decomposition of primes in algebraic number fields

Abstract

It is shown that factoring polynomials over finite prime fields is polynomial-time equivalent to decomposing primes in algebraic number fields whose generating polynomials have discriminants not divisible by the given primes. The reduction from polynomial factorization to prime decomposition suggests a number-theoretic approach to the former problem. Along this line, two results will be shown based on the generalized Riemann hypothesis (GRH): 1. (1) Given p, n ϵ Z >0 with p prime, all the solutions to Φ n ( x) = O( p) can be found in time polynomial in n and log p, where Φ n denotes the nth cyclotomic polynomial. 2. (2) Given p, n, a ϵ Z >0 with p prime, all the solutions to x n = a( p) can be found in time polynomial in n, log p, and log a.