Infinite families of non-monogenic trinomials. (English) Zbl 1488.11163
Summary: Let \(f(x) \in \mathbb{Z}[x]\) be monic and irreducible over \(\mathbb{Q}\), with \(\deg (f) = n\). Let \(K = \mathbb{Q}(\vartheta)\), where \(f(\vartheta) = 0\), and let \(\mathbb{Z}_K\) denote the ring of integers of \(K\). We say \(f(x)\) is non-monogenic if \(\{1,\vartheta,\vartheta^2,\ldots,\vartheta^{n-1}\}\) is not a basis for \(\mathbb{Z}_K\). By extending ideas of L. J. Ratliff jun. et al. [J. Number Theory 121, No. 1, 90–113 (2006; Zbl 1171.12002)] we construct infinite families of non-monogenic trinomials.
MSC:
| 11R04 | Algebraic numbers; rings of algebraic integers |
| 11R09 | Polynomials (irreducibility, etc.) |
| 12F05 | Algebraic field extensions |
Citations:
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