On common index divisors and monogenity of certain number fields defined by trinomials of type $x^{2^r}+ax^m+b$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913500843999232 |
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| author | Yakkou, Hamid Ben |
| author_facet | Yakkou, Hamid Ben |
| contents | Let $K = \Q(þ)$ be a number with $þ$ a root of an irreducible trinomial of type $ F(x)= x^{2^r}+ax^m+b \in \Z[x]$. In this paper, based on the $p$-adic Newton polygon techniques applied on decomposition of primes in number fields and the classical index theorem of Ore \cite{Narprime, O}, we study the monogenity of $K$. More precisely, we prove that if $a$ and $1+b$ are both divisible by $32$, then $K$ cannot be monogenic. For $m=1$, we provide explicit conditions on $a$, $b$ and $r$ for which $K$ is not monogenic. We also construct a family of irreducible trinomials which are not monogenic, but their roots generate monogenic number fields. To illustrate our results, we give some computational examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_10413 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On common index divisors and monogenity of certain number fields defined by trinomials of type $x^{2^r}+ax^m+b$ Yakkou, Hamid Ben Number Theory 11R04, 11R21, 11R16, 11Y04 Let $K = \Q(þ)$ be a number with $þ$ a root of an irreducible trinomial of type $ F(x)= x^{2^r}+ax^m+b \in \Z[x]$. In this paper, based on the $p$-adic Newton polygon techniques applied on decomposition of primes in number fields and the classical index theorem of Ore \cite{Narprime, O}, we study the monogenity of $K$. More precisely, we prove that if $a$ and $1+b$ are both divisible by $32$, then $K$ cannot be monogenic. For $m=1$, we provide explicit conditions on $a$, $b$ and $r$ for which $K$ is not monogenic. We also construct a family of irreducible trinomials which are not monogenic, but their roots generate monogenic number fields. To illustrate our results, we give some computational examples. |
| title | On common index divisors and monogenity of certain number fields defined by trinomials of type $x^{2^r}+ax^m+b$ |
| topic | Number Theory 11R04, 11R21, 11R16, 11Y04 |
| url | https://arxiv.org/abs/2203.10413 |