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- W4376988433 abstract "Let $L_n$ be the simplest cubic field with the Galois group $G=mathrm{Gal}(L_n/mathbb Q)$. The associated order is denoted as ${mathcal A}_{L_n/mathbb Q}:= { xin mathbb Q [G] , |, x. {mathcal O}_{L_n} subset {mathcal O}_{L_n } }$, where ${mathcal O}_{L_n}$ is the ring of integers of $L_n$. Leopoldt showed that ${mathcal O}_{L_n} simeq {mathcal A}_{L_n/mathbb Q}$ as ${mathcal A}_{L_n/mathbb Q}$-modules. In this paper, we give the generator of the ${mathcal A}_{L_n/mathbb Q}$-module ${mathcal O}_{L_n}$ explicitly using the roots of Shanks' cubic polynomial. If $L_n/mathbb Q$ is tamely ramified, then we have ${mathcal A}_{L_n/mathbb Q}=mathbb Z [G]$, and the conjugates of the generator is a normal integral basis, which has been obtained explicitly in the previous work of Hashimoto and Aoki." @default.
- W4376988433 created "2023-05-18" @default.
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- W4376988433 date "2023-05-15" @default.
- W4376988433 modified "2023-09-29" @default.
- W4376988433 title "Galois module structure of algebraic integers of the simplest cubic field" @default.
- W4376988433 doi "https://doi.org/10.48550/arxiv.2305.08888" @default.
- W4376988433 hasPublicationYear "2023" @default.
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