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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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2025-07-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | https://mb.math.cas.cz/full/150/2/mb150_2_5.pdf |
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| 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. |
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| ISSN: | 0862-7959 2464-7136 |