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Non-monogenity in a family of octic fields. (English) Zbl 1381.11102

The ring of integers \({\mathbb Z}_{K}\) of the number field \(K\) of degree \(d\) over \({\mathbb Q}\) is called monogene if there is \(\alpha \in {\mathbb Z}_{K}\) such that \(1,\alpha,\dots,\alpha^{n-1}\) is an integral basis of \(K\). The authors prove that for each square-free integer \(m \equiv 2,3 \pmod 4\) the octic field \(K={\mathbb Q}(i, \sqrt[4]{m})\) is not monogene (Theorem 1.1). The proof involves some calculations.

MSC:

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