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On integral bases of certain real octic abelian fields. (English) Zbl 1158.11342

Summary: Let \(K=\mathbb Q(\sqrt{mn}, \sqrt{mn}, \sqrt{d_1m_1n_1\ell})\) be an octic field. If \(K\) is monogenic and a quadratic subfield \(\mathbb Q(\sqrt{d_1m_1n_1})\) of \(K\) and a quartic subfield \(\mathbb Q(\sqrt{mn},\sqrt{dn})\) are linearly disjoint, then \(K\) coincides with the field \(\mathbb Q(\sqrt{-1},\sqrt{2},\sqrt{-3})\); namely \(K\) is equal to the cyclotomic field \(\mathbb Q(\zeta_{24})\) [Y. Motoda and T. Nakahara, Arch. Math. 83, No. 4, 309–316 (2004; Zbl 1078.11061)]. In this paper, we determine an explicit integral basis of an octic field. Next, we prove that certain real octic fields \(K\) are non-monogenic slightly, but essentially generalizing the conditions in Theorem 0 (loc. cit.).

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R20 Other abelian and metabelian extensions

Citations:

Zbl 1078.11061
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