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On common index divisor of the number fields defined by $x^7+ax+b$

On common index divisor of the number fields defined by $x^7+ax+b$

Authors :
Jakhar, Anuj
Kaur, Sumandeep
Kumar, Surender
Publication Year :
2023

Abstract

Let $f(x)=x^7+ax+b$ be an irreducible polynomial having integer coefficients and $K=\mathbb{Q}(\theta)$ be an algebraic number field generated by a root $\theta$ of $f(x)$. In the present paper, for every rational prime $p$, our objective is to determine the necessary and sufficient conditions involving only $a,~b$ so that $p$ is a divisor of the index of the field $K$. In particular, we provide sufficient conditions on $a$ and $b$, for which $K$ is non-monogenic. In a special case, we show that if either $8$ divides both $a\pm1$, $b$ or $32$ divides both $a+4$, $b$, then $K$ is non-monogenic. We illustrate our results through examples.<br />Comment: arXiv admin note: text overlap with arXiv:2011.14348

Subjects

Subjects :
Mathematics - Number Theory

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2301.00365
Document Type :
Working Paper