On monogenity of certain number fields defined by trinomials. (English) Zbl 1515.11108

A monic irreducible polynomial \(f(x)\) belonging to \(\mathbb{Z}[x]\) having a root \(\theta\) in \(\mathbb{C}\) is said to be monogenic if \(\mathbb{Z}[\theta]\) is the ring of algebraic integers of the algebraic number field \(\mathbb{Q}(\theta)\). An algebraic number field \(K\) with ring of algebraic integers denoted by \(A_{K}\) is said to be monogenic if there exists \(\alpha \in A_{K}\) such that \(A_{K}=\mathbb{Z}[\alpha]\). In this paper, the authors mainly deal with the problem of constructing families of non-monogenic fields which are generated over \(\mathbb{Q}\) by some root of an irreducible trinomial of the type \(f(x)=x^n+ax+b\) belongs to \(\mathbb{Z}[x]\). Examples are also given when a trinomial \(f(x)\) having a root \(\theta\) in \(\mathbb{C}\) is not monogenic but the field \(\mathbb{Q}(\theta)\) is monogenic. The authors use theorem of index of Ore.


11R04 Algebraic numbers; rings of algebraic integers
11R21 Other number fields
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