Abstract
In this paper, we study the monogenity of any number field defined by a monic irreducible trinomial \(F(x)=x^{12}+ax^m+b\in \mathbb {Z}[x]\) with \(1\le m\le 11\) an integer. For every integer m, we give sufficient conditions on a and b so that the field index i(K) is not trivial. In particular, if \(i(K)\ne 1\), then K is not monogenic. For \(m=1\), we give necessary and sufficient conditions on a and b, which characterize when a rational prime p divides the index i(K). For every prime divisor p of i(K), we also calculate the highest power p dividing i(K), in such a way we answer the problem 22 of Narkiewicz (Elementary and analytic theory of algebraic numbers, Springer Verlag, Auflag, 2004) for the number fields defined by trinomials \(x^{12}+ax+b\).
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Acknowledgements
The authors are very grateful to the anonymous referee for his careful reading of the paper. The first author is very grateful to Professor István Gaál for his advice and encouragement as well as to Professor Enric Nart who introduced him to Newton polygon techniques.
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This paper is dedicated to István Gaál.
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El Fadil, L., Kchit, O. On index divisors and monogenity of certain number fields defined by \(x^{12}+ax^m+b\). Ramanujan J 63, 451–482 (2024). https://doi.org/10.1007/s11139-023-00768-4
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DOI: https://doi.org/10.1007/s11139-023-00768-4
Keywords
- Theorem of Dedekind
- Theorem of Ore
- Prime ideal factorization
- Newton polygon
- Index of a number field
- Power integral basis
- Monogenic