Startseite Mathematik On index divisors of certain sextic number fields defined by quadrinomials
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On index divisors of certain sextic number fields defined by quadrinomials

  • Hamid Ben Yakkou , Issam Aghzer und Abdelkarim Boua EMAIL logo
Veröffentlicht/Copyright: 11. Oktober 2025
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Abstract

Let K be a number field of degree 6 generated by a root of an irreducible quadrinomial F ( x ) = x 6 + a x m + b x 2 + c [ x ] with m { 3 , 4 , 5 } , and let i ( K ) denote the index of K. In this paper, for p = 2 or 3, we give sufficient conditions on a , b and c such that p is a common index divisor of K, and we evaluate ν p ( i ( K ) ) , the highest power of p dividing i ( K ) . In this way, we provide a partial answer to Problem 22 of [W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer Monogr. Math., Springer, Berlin, 2004] for these families of number fields. As an application of our results, we identify new classes of sextic number fields having no power integral basis. Further, several corollaries and examples are provided which illustrate our results. Our approach is based on a theorem of Ore on the decomposition of primes in number fields. [18, 30, 34].

Acknowledgements

The authors are deeply grateful to the referee, whose valuable comments and suggestions have greatly improved the quality of this paper.

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Received: 2025-02-26
Revised: 2025-08-23
Accepted: 2025-08-27
Published Online: 2025-10-11

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