| Title:
|
On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$ (English) |
| Author:
|
Ben Yakkou, Hamid |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
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0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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150 |
| Issue:
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2 |
| Year:
|
2025 |
| Pages:
|
245-262 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
monogenity |
| Keyword:
|
power integral basis |
| Keyword:
|
theorem of Ore |
| Keyword:
|
prime ideal factorization |
| Keyword:
|
common index divisor |
| Keyword:
|
Newton polygon |
| MSC:
|
11R04 |
| MSC:
|
11R16 |
| MSC:
|
11R21 |
| MSC:
|
11Y40 |
| DOI:
|
10.21136/MB.2024.0148-23 |
| . |
| Date available:
|
2025-05-20T11:56:34Z |
| Last updated:
|
2025-05-20 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152974 |
| . |
| Reference:
|
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