On power basis of a class of number fields

2023/03/06 by Jakhar, Anuj, Kaur, Sumandeep, Kumar, Surender
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2303.03138

Abstract

Let f(x)=xⁿ+ax²+bx+c ∈ \Z[x] be an irreducible polynomial with b²=4ac and let K=\Q(θ) be an algebraic number field defined by a complex root θ of f(x). Let \Z_K deonote the ring of algebraic integers of K. The aim of this paper is to provide the necessary and sufficient conditions involving only a,c and n for a given prime p to divide the index of the subgroup \Z[θ] in \Z_K. As a consequence, we provide families of monogenic algebraic number fields. Further, when \Z_K ≠ \Z[θ], we determine explicitly the index [\Z_K : \Z[θ]] in some cases.

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