On common index divisors and monogenity of of the nonic number field defined by a trinomial $x^9+ax+b$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913504798179328 |
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| author | Yakkou, Hamid Ben Tiebekabe, Pagdame |
| author_facet | Yakkou, Hamid Ben Tiebekabe, Pagdame |
| contents | Let $K $ be a nonic number field generated by a complex root $þ$ of a monic irreducible trinomial $ F(x)= x^9+ax+b \in \Z[x]$, where $ab \neq 0$. Let $i(K)$ be the index of $K$. A rational prime $p$ dividing $ i(K)$ is called a prime common index divisor of $K$. In this paper, for every rational prime $p$, we give necessary and sufficient conditions depending only $a$ and $b$ for which $p$ is a common index divisor of $K$. As application of our results we identify infinite parametric families of non-monogenic nonic numbers fields defined by such trinomials. At the end, some numerical examples illustrating our theoretical results are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_05029 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On common index divisors and monogenity of of the nonic number field defined by a trinomial $x^9+ax+b$ Yakkou, Hamid Ben Tiebekabe, Pagdame Number Theory 11R16, 11R04 F.2.2 Let $K $ be a nonic number field generated by a complex root $þ$ of a monic irreducible trinomial $ F(x)= x^9+ax+b \in \Z[x]$, where $ab \neq 0$. Let $i(K)$ be the index of $K$. A rational prime $p$ dividing $ i(K)$ is called a prime common index divisor of $K$. In this paper, for every rational prime $p$, we give necessary and sufficient conditions depending only $a$ and $b$ for which $p$ is a common index divisor of $K$. As application of our results we identify infinite parametric families of non-monogenic nonic numbers fields defined by such trinomials. At the end, some numerical examples illustrating our theoretical results are given. |
| title | On common index divisors and monogenity of of the nonic number field defined by a trinomial $x^9+ax+b$ |
| topic | Number Theory 11R16, 11R04 F.2.2 |
| url | https://arxiv.org/abs/2212.05029 |