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 Φn(x) = O(p) can be found in time polynomial in n and log p, where Φn denotes the nth cyclotomic polynomial. 2. (2) Given p,n,a ϵ Z>0 with p prime, all the solutions to xn = a(p) can be found in time polynomial in n, log p, and log a.