Rocky Mountain Journal of Mathematics

Non-monogenity in a family of octic fields

István Gaál and László Remete

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Let $m$ be a square-free integer, $m\equiv 2,3\pmod 4$. We show that the number field $K=\mathbb{Q} (i,\sqrt [4]{m})$ is non-monogene, that is, it does not admit any power integral bases of type $\{1,\alpha ,\ldots ,\alpha ^7\}$. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using congruence considerations only.

Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields which is applicable for parametric families of number fields. We calculate the index of elements as polynomials dependent upon the parameter, factor these polynomials, and consider systems of congruences according to the factors.

Article information

Rocky Mountain J. Math., Volume 47, Number 3 (2017), 817-824.

First available in Project Euclid: 24 June 2017

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Zentralblatt MATH identifier

Primary: 11R04: Algebraic numbers; rings of algebraic integers 11Y50: Computer solution of Diophantine equations

Power integral basis octic fields relative quartic extension


Gaál, István; Remete, László. Non-monogenity in a family of octic fields. Rocky Mountain J. Math. 47 (2017), no. 3, 817--824. doi:10.1216/RMJ-2017-47-3-817.

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