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arXiv · 1809.10084

Integral bases and monogenity of pure fields

István Gaál·László Remete
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Abstract

Let $m$ be a square-free integer ($m\neq 0,\pm 1$). We show that the structure of the integral bases of the fields $K=Q(\sqrt[n]{m})$ are periodic in $m$. For $3\leq n\leq 9$ 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.

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István Gaál, László Remete. 2018-09-26. Integral bases and monogenity of pure fields. https://doi.org/10.1016/j.jnt.2016.09.009

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