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On index divisors of septic number fields defined by $x^7+ax^2+bx+c$ | ||
| Journal of Algebra and Related Topics | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 26 مرداد 1405 اصل مقاله (217.56 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2026.31130.1823 | ||
| نویسندگان | ||
| Ben Yakkou H.* 1؛ A. Boua2؛ M. Rabiy2 | ||
| 1Polydisciplinary Faculty of Béni-Mellal, University Sultan Moulay Slimane, Béni-Mellal, Morocco | ||
| 2Department of Mathematics, Polydisciplinary Faculty, University Sidi Mohammed Ben Abdellah, Taza, Morocco | ||
| چکیده | ||
| Consider a septic number field $K$ generated by a root of an irreducible quadrinomial $F(x)= x^7+ax^2+bx+c \in \mathbb{Z} [x]$. Let $i(K)$ denote the index of $K$, which is the greatest common divisor of indices of all primitive elements of the ring of integers of $K$. In this paper, we show that $2$ and $3$ are the only primes that can divide $i(K)$. Furthermore, for $p=2$ and $3$, we give sufficient conditions on $a, b$ and $c$ for which $i(K)$ is divisible by $p$ and we determine $\nu_p(i(K))$, the highest power of $p$ dividing $i(K)$. In particular, if $i(K)\neq 1$, then $K$ cannot be monogenic. Our results provide a partial answer to Problem $22$ of Narkiewicz [Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, Berlin, Heidelberg (2004)] for these families of number fields. | ||
| کلیدواژهها | ||
| Index of a number field؛ Newton polygon؛ Theorem of Ore؛ Prime ideal factorization؛ Monogenity | ||
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آمار تعداد مشاهده مقاله: 5 تعداد دریافت فایل اصل مقاله: 3 |
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