On common index divisor of the number fields defined by x⁷+ax+b

2023/01/01 by Anuj Jakhar, Sumandeep Kaur, Jakhar, Anuj +3
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.00365

openalex publication_date 2023/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f(x)=x⁷+ax+b be an irreducible polynomial having integer coefficients and K=ℚ(θ) be an algebraic number field generated by a root θ 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±1, b or 32 divides both a+4, b, then K is non-monogenic. We illustrate our results through examples.

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