On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$
Let $K $ be a septic number field generated by a root $\theta$ of an irreducible polynomial $ F(x)= x^7+ax^5+b \in\mathbb Z[x]$. In this paper, we explicitly characterize the index $i(K)$ of $K$. More precisely, for all $a$ and $b$, we show that $i(K) \in\{1, 2\}$. Our results answer completely to P...
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Institute of Mathematics of the Czech Academy of Science
2025-07-01
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| Series: | Mathematica Bohemica |
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| Online Access: | https://mb.math.cas.cz/full/150/2/mb150_2_5.pdf |
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| author | Hamid Ben Yakkou |
| author_facet | Hamid Ben Yakkou |
| author_sort | Hamid Ben Yakkou |
| collection | DOAJ |
| description | Let $K $ be a septic number field generated by a root $\theta$ of an irreducible polynomial $ F(x)= x^7+ax^5+b \in\mathbb Z[x]$. In this paper, we explicitly characterize the index $i(K)$ of $K$. More precisely, for all $a$ and $b$, we show that $i(K) \in\{1, 2\}$. Our results answer completely to Problem 22 of W. Narkiewicz's book (2004) for these families of number fields. In particular, we provide sufficient conditions for which $K$ is not monogenic. We illustrate our results by some computational examples. |
| format | Article |
| id | doaj-art-8745e6a220944089ab6f2f4e1b9c3702 |
| institution | DOAJ |
| issn | 0862-7959 2464-7136 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj-art-8745e6a220944089ab6f2f4e1b9c37022025-08-20T02:58:22ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362025-07-01150224526210.21136/MB.2024.0148-23MB.2024.0148-23On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$Hamid Ben YakkouLet $K $ be a septic number field generated by a root $\theta$ of an irreducible polynomial $ F(x)= x^7+ax^5+b \in\mathbb Z[x]$. In this paper, we explicitly characterize the index $i(K)$ of $K$. More precisely, for all $a$ and $b$, we show that $i(K) \in\{1, 2\}$. Our results answer completely to Problem 22 of W. Narkiewicz's book (2004) for these families of number fields. In particular, we provide sufficient conditions for which $K$ is not monogenic. We illustrate our results by some computational examples.https://mb.math.cas.cz/full/150/2/mb150_2_5.pdf monogenity power integral basis theorem of ore prime ideal factorization common index divisor newton polygon |
| spellingShingle | Hamid Ben Yakkou On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$ Mathematica Bohemica monogenity power integral basis theorem of ore prime ideal factorization common index divisor newton polygon |
| title | On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$ |
| title_full | On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$ |
| title_fullStr | On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$ |
| title_full_unstemmed | On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$ |
| title_short | On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$ |
| title_sort | on common index divisors and monogenity of septic number fields defined by trinomials of type x 7 ax 5 b |
| topic | monogenity power integral basis theorem of ore prime ideal factorization common index divisor newton polygon |
| url | https://mb.math.cas.cz/full/150/2/mb150_2_5.pdf |
| work_keys_str_mv | AT hamidbenyakkou oncommonindexdivisorsandmonogenityofsepticnumberfieldsdefinedbytrinomialsoftypex7ax5b |