On indices and monogenity of quartic number fields defined by quadrinomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916380592308224 |
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| author | Yakkou, Hamid Ben |
| author_facet | Yakkou, Hamid Ben |
| contents | Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $ F(x)= x^4+ax^3+bx+c \in \Z[x]$. Let $i(K)$ denote the index of $K$. Engstrom \cite{Engstrom} established that $i(K)=2^u \cdot 3^v$ with $u \le 2$ and $v \le 1$. In this paper, we provide sufficient conditions on $a$, $b$ and $c$ for $i(K)$ to be divisible by $2$ or $3$, determining the exact corresponding values of $u$ and $v$ in each case.
In particular, when $i(K) \neq 1$, $K$ cannot be monogenic. We also identify new infinite parametric families of monogenic quartic number fields generated by roots of non-monogenic quadrinomials. We illustrate our results by some computational examples. Our method is based on a theorem of Ore on the decomposition of primes in number fields \cite{Nar,O}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_12782 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On indices and monogenity of quartic number fields defined by quadrinomials Yakkou, Hamid Ben Number Theory 11R04, 11R16, 11R21, 11Y40 F.2.2 Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $ F(x)= x^4+ax^3+bx+c \in \Z[x]$. Let $i(K)$ denote the index of $K$. Engstrom \cite{Engstrom} established that $i(K)=2^u \cdot 3^v$ with $u \le 2$ and $v \le 1$. In this paper, we provide sufficient conditions on $a$, $b$ and $c$ for $i(K)$ to be divisible by $2$ or $3$, determining the exact corresponding values of $u$ and $v$ in each case. In particular, when $i(K) \neq 1$, $K$ cannot be monogenic. We also identify new infinite parametric families of monogenic quartic number fields generated by roots of non-monogenic quadrinomials. We illustrate our results by some computational examples. Our method is based on a theorem of Ore on the decomposition of primes in number fields \cite{Nar,O}. |
| title | On indices and monogenity of quartic number fields defined by quadrinomials |
| topic | Number Theory 11R04, 11R16, 11R21, 11Y40 F.2.2 |
| url | https://arxiv.org/abs/2401.12782 |