Gauss periods are minimal polynomials for totally real cyclic fields of prime degree

TL;DR

This paper provides extensive computational evidence that Gauss period equations serve as minimal polynomials for primitive elements in totally real cyclic fields of prime degree, extending known tables and conjecturing minimality in many cases.

Contribution

It significantly extends the database of minimal polynomials for prime degree cyclic fields using Gauss period equations, including new conjectures for minimal discriminants.

Findings

Computed 200 period equations up to p=97, extending existing tables.

Confirmed known minimal discriminant cases for p ≤ 7.

Identified 128 new cases of minimal discriminant polynomials for 29 ≤ p ≤ 97.

Abstract

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 Kl\"uners and Malle. Up to $p=7$, period equations reproduce known results proved to have minimum discriminant. For $11\leq p\leq 23$, period equations coincide with 53 known but unproved cases of minimum discriminant in the database, and fill a gap of 19 missing cases. For $29\leq p\leq 97$, 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.