Galois module structure of algebraic integers of cyclic cubic fields
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910028362940416 |
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| author | Aoki, Miho |
| author_facet | Aoki, Miho |
| contents | We determine the Galois module structure of the ring of integers for all cubic fields using roots of the generic cyclic cubic polynomial $f_n(X)=X^3-nX^2-(n+3)X-1$. 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 ${\mathcal O}_{L_n}$ the ring of integers of $L_n$. We explicitly give the generator of the free module ${\mathcal O}_{L_n}$ of rank $1$ over the associated order ${\mathcal A}_{L_n/\mathbb Q}:= \{ x\in \mathbb Q [G] \, |\, x\, {\mathcal O}_{L_n} \subset {\mathcal O}_{L_n} \}$ by using the roots of $f_n(X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20403 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Galois module structure of algebraic integers of cyclic cubic fields Aoki, Miho Number Theory 11R04, 11R16, 11C08, 11R80 We determine the Galois module structure of the ring of integers for all cubic fields using roots of the generic cyclic cubic polynomial $f_n(X)=X^3-nX^2-(n+3)X-1$. 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 ${\mathcal O}_{L_n}$ the ring of integers of $L_n$. We explicitly give the generator of the free module ${\mathcal O}_{L_n}$ of rank $1$ over the associated order ${\mathcal A}_{L_n/\mathbb Q}:= \{ x\in \mathbb Q [G] \, |\, x\, {\mathcal O}_{L_n} \subset {\mathcal O}_{L_n} \}$ by using the roots of $f_n(X)$. |
| title | Galois module structure of algebraic integers of cyclic cubic fields |
| topic | Number Theory 11R04, 11R16, 11C08, 11R80 |
| url | https://arxiv.org/abs/2410.20403 |