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arXiv · 2212.06115

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

Jason A. C. Gallas
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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.

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Jason A. C. Gallas. 2022-12-06. Gauss periods are minimal polynomials for totally real cyclic fields of prime degree. https://arxiv.org/abs/2212.06115

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