Galois module structure of algebraic integers of cyclic cubic fields

2024/10/27 by Miho Aoki, Aoki, Miho · 1 citation
Computer Science · Mathematics · #11C08 #11R04 #11R16 #11R80 #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2410.20403

openalex publication_date 2024/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We determine the Galois module structure of the ring of integers for all cubic fields using roots of the generic cyclic cubic polynomial f_n(X)=X³-nX²-(n+3)X-1. Let L_n=\mathbb Q(ρ_n) be a cyclic cubic field with Galois group G:=\rm Gal(L_n/\mathbb Q), where ρ_n is a root of f_n (X), and \mathcal O_Ln the ring of integers of L_n. We explicitly give the generator of the free module \mathcal O_Ln of rank 1 over the associated order \mathcal A_{Ln/\mathbb Q}:= \ x∈ \mathbb Q [G] | x \mathcal O_Ln ⊂ \mathcal O_Ln \ by using the roots of f_n(X).

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• Gaussian periods and Shanks' cubic polynomials. II

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