2022/12/06 by Jason A. C. Gallas, Gallas, Jason A. C.
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2212.06115
openalex publication_date 2022/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We report extensive computational evidence that Gauss period equations are minimal discriminant polynomials for primitive elements representing Abelian (cyclic) polynomials of prime degrees p. By computing 200 period equations up to p=97, we significantly extend tables in the compendious number fields database of Klüners and Malle. Up to p=7, period equations reproduce known results proved to have minimum discriminant. For 11≤ p≤ 23, period equations coincide with 53 known but unproved cases of minimum discriminant in the database, and fill a gap of 19 missing cases. For 29≤ p≤ 97, we report 128 not previously known cases, 16 of them conjectured to be minimum discriminant polynomials of Galois group pT1. The significant advantage of period equations is that they all may be obtained analytically using a procedure that works for fields of arbitrary degrees, and which are extremely hard to detect by systematic numerical search.