On common index divisors and monogenity of certain number fields defined by $x^5+ax+b$

Lhoussain El Fadil

On common index divisors and monogenity of certain number fields defined by $x^5+ax+b$

Number Theory 2023-06-21 v3

Authors: Lhoussain El Fadil

Abstract

The goal of this paper is to calculate explicitly the field index of any quintic number field $K$ generated by a complex root $\al$ of a monic irreducible trinomial $F(x) = x^5+ax+b \in \Z[x]$ . In such a way we provide a complete answer to the Problem 22 of Narkiewicz \cite{WN}. Namely for every prime integer $p$ , we evaluate the highest power of $p$ dividing $i(K)$ . In particular, we give sufficient conditions on $a$ and $b$ , which guarantee the non monogenity of $K$ .

Keywords

quintic number fields field index common index divisors monogenity trinomials

Cite

@article{arxiv.2112.01133,
  title  = {On common index divisors and monogenity of certain number fields defined by $x^5+ax+b$},
  author = {Lhoussain El Fadil},
  journal= {arXiv preprint arXiv:2112.01133},
  year   = {2023}
}

Comments

Submitted. arXiv admin note: text overlap with arXiv:2111.05899, arXiv:2106.01252, arXiv:2109.08765

On common index divisors and monogenity of certain number fields defined by $x^5+ax+b$

Abstract

Keywords

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