A HYBRID GROBNER BASES APPROACH TO COMPUTING POWER INTEGRAL BASES

摘要

Bettale, Faugere, and Perret [3] present and analyze a hybrid method for solving multivariate polynomial systems over finite fields that mixes Grobner bases computations with an exhaustive search. Inspired by their method, we use a hybrid approach to characterize all power integral bases in the pth cyclotomic field Q(zeta(p)) for the regular primes p = 29, 31, 41. For each prime p this involves solving a system of (p-1)/2 multivariate polynomial equations of degree (p - 1)/2 in (p - 1)/2 variables over the finite field Z/pZ.