Galois module structure of algebraic integers of the simplest cubic field

2023/05/15 by Ogawa, Hajime, Aoki, Miho
#11R16 #11R80 #FOS: Mathematics #Number Theory (math.NT) #Primary 11R04 #Secondary 11C08

paper · doi:10.48550/arxiv.2305.08888

Abstract

Let L_n be a simplest cubic field with Galois group G=\rmGal (L_n/\mathbb Q). The associated order is denoted as \cal A_{Ln/\mathbb Q}:= \ x∈ \mathbb Q [G] | x ⋅ \calO_Ln ⊂ \cal O_Ln \, where \cal O_Ln is the ring of integers of L_n. Leopoldt showed that \calO_Ln ≃ \cal A_{Ln/\mathbb Q} as \cal A_{Ln/\mathbb Q}-modules. In this paper, we give a generator of the \cal A_{Ln/\mathbb Q}-module \cal O_Ln explicitly using the roots of Shanks' cubic polynomial. If L_n/\mathbb Q is tamely ramified, then we have \cal A_{Ln/\mathbb Q}=\mathbb Z [G], and the conjugates form a normal integral basis, which has been obtained explicitly in the previous work of Hashimoto and the second author.

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