Integral bases and monogenity of pure fields

Abstract

Let $m$ be a square-free integer ($m\ne 0,\pm 1$). We show that the structure of the integral bases of the fields $K=Q(\sqrt[n]{m})$ are periodic in $m$. For $3\le n\le 9$ we show that the period length is $n^2$. We explicitly describe the integral bases, and for $n=3,4,5,6,8$ we explicitly calculate the index forms of $K$. This enables us in many cases to characterize the monogenity of these fields. Using the explicit form of the index forms yields a new technic that enables us to derive new results on monogenity and to get several former results as easy consequences. For $n=4,6,8$ we give an almost complete characterization of the monogenity of pure fields.

References in corpus (1)

• Binomial Thue equations and power integral bases in pure quartic fields

Cited by in corpus (7)

• On non-monogenic number fields defined by trinomials of type $x^n+a{x}^m+b$
• Integral basis of pure fields with square-free parameter
• On power integral bases of certain pure number fields defined by $x^{2^u\cdot 3^v}-m$
• On common index divisors and monogenity of of the nonic number field defined by a trinomial $x^9+ax+b$
• On common index divisors and monogenity of certain number fields defined by $x^5+ax+b$
• A generalization of simplest number fields and their integral basis
• The Monogeneity of Kummer Extensions and Radical Extensions