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arXiv · 2304.00339

DISCRIMINANT and integral basis OF $\mathbb{Q}(\sqrt[12]{a})$

Anuj Jakhar·Surender Kumar
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Abstract

Suppose $m$ be a $12$-th power free integer. Let $K=\mathbb{Q}(\theta)$ be an algebraic number field defined by a complex root $\theta$ of an irreducible polynomial $x^{12}-m$ and $O_K$ be its ring of integers. In this paper, we determine the highest power of $p$ dividing the index of the subgroup $\Z[\theta]$ in $O_K$ and $p$-integral basis of $K$ for each prime $p$. These $p$-integral bases lead to the construction of an integral basis of $K$ which is illustrated with examples. In particular, when $m$ is a square free integer, we provide necessary and sufficient conditions for the set $\{1,\theta,\theta^2,\cdots,\theta^{10},\theta^{11}\}$ to be an integral basis of $K.$

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Anuj Jakhar, Surender Kumar. 2023-04-01. DISCRIMINANT and integral basis OF $\mathbb{Q}(\sqrt[12]{a})$. https://arxiv.org/abs/2304.00339

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