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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2410.17560 |
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| _version_ | 1866914985429434368 |
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| author | Venkataraman, S. Kulkarni, Manisha V. |
| author_facet | Venkataraman, S. Kulkarni, Manisha V. |
| contents | Let $K=\mathbb{Q}[ι]$ and $N=K[\sqrt[4]α]$, $α\in\mathbb{Z}[ι]$, $alpha=fg^2h^3$, $f$, $g$, $h\in \mathbb{Z}[ι]$ are pairwise coprime and square free. Let $\mathcal{O}_N$ be the ring of integers of $N$. In this article we construct normalised integral basis for $\mathcal{O}_N$ over $\mathbb{Z}[ι]$, that is an integral basis of the form \[ \left\{1,\frac{f_1(α)}{d_1},\frac{f_2(α)}{d_2},\frac{f_{3}(α)}{d_3}\right\} \] where $d_i \in \mathbb{Z}[i]$ and $f_i(X)$, $\leq i\leq 3$ are monic polynomials of degree $i$ over $\mathbb{Z}[ι]$. We explicitly determine what $d_i$, $1\leq i\leq n-1$ are in terms of $f$, $g$ and $h$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral Basis for quartic Kummer extensions over $\mathbb{Z}[ι]$ Venkataraman, S. Kulkarni, Manisha V. Number Theory Let $K=\mathbb{Q}[ι]$ and $N=K[\sqrt[4]α]$, $α\in\mathbb{Z}[ι]$, $alpha=fg^2h^3$, $f$, $g$, $h\in \mathbb{Z}[ι]$ are pairwise coprime and square free. Let $\mathcal{O}_N$ be the ring of integers of $N$. In this article we construct normalised integral basis for $\mathcal{O}_N$ over $\mathbb{Z}[ι]$, that is an integral basis of the form \[ \left\{1,\frac{f_1(α)}{d_1},\frac{f_2(α)}{d_2},\frac{f_{3}(α)}{d_3}\right\} \] where $d_i \in \mathbb{Z}[i]$ and $f_i(X)$, $\leq i\leq 3$ are monic polynomials of degree $i$ over $\mathbb{Z}[ι]$. We explicitly determine what $d_i$, $1\leq i\leq n-1$ are in terms of $f$, $g$ and $h$. |
| title | Integral Basis for quartic Kummer extensions over $\mathbb{Z}[ι]$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.17560 |