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On nonmonogenic number fields defined by trinomials of type \(x^n +ax^m+b\). (English) Zbl 07731139

Let \(f(X) \in \mathbb{Z}[X]\) be an irreducible polynomial. Let \(\theta \) be a root of \(f\) and let \(K=\mathbb{Q} (\theta)\). The author calls the polynomial \(f\) monogenic if the ring of integers of \(K\) is of the form \(\mathbb{Z} [\theta]\), in other words, a root of \(f\) generates a power integral basis of \(K\). For a large number of references on this subject, refer to the introduction of this paper.
The aim of this paper under review is to provide sufficient conditions for a trinomial of the form \(f(X)=X^n + a X^m +b \in \mathbb{Z}[X]\) to be non-monogenic. The author uses the Newton polygon method to enumerate the irreducible factors of \(f\) modulo a prime \(p\) and finds a condition for \(p\) to be a divisor of the common index divisor, thus indicating that it is not monogenic.
The main results obtained are too complicated to state here due to the many case-by-case descriptions involved. By applying the main results, the author obtained an infinite subfamily of trinomials of degree \(2^k 3^r \) for which \(3\) divides the common index divisor.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R16 Cubic and quartic extensions
11R21 Other number fields
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References:

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