On index and monogenity of certain number fields defined by trinomials. (English) Zbl 1524.11190
Summary: Let \(K\) be a number field generated by a root \(\theta\) of a monic irreducible trinomial \(F(x)=x^n+ax^m+b\in\mathbb{Z}[x]\). In this paper, we study the problem of monogenity of \(K\). More precisely, we provide some explicit conditions on \(a\), \(b\), \(n\), and \(m\) for which \(K\) is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree \(n=2^r\cdot 3^k\) with \(r\) and \(k\) two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this paper.
MSC:
| 11R04 | Algebraic numbers; rings of algebraic integers |
| 11R21 | Other number fields |
| 11Y40 | Algebraic number theory computations |
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