Abstract
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 \({\mathbb{Q}(\zeta_p)}\) 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 \({\mathbb{Z}/p\mathbb{Z}}\).
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