On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$
UDC 511 For any octic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^8+ax^3+b \in \mathbb{Z}[x]$ and for every rational prime $p,$ we show when $p$ divides the index of $K.$  We also describe the prime power decomposition of the index...
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| Date: | 2024 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7536 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 511
For any octic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^8+ax^3+b \in \mathbb{Z}[x]$ and for every rational prime $p,$ we show when $p$ divides the index of $K.$  We also describe the prime power decomposition of the index $i(K).$ In this way, we give a partial answer to { Problem $22$} of Narkiewicz [Elementary and analytic theory of algebraic numbers, Springer-Verlag, Auflage (2004)] for this family of number fields. As an application of our results, we conclude that if $i(K)\neq1,$  then $K$ is not monogenic. We illustrate our results by some computational examples.  |
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| DOI: | 10.3842/umzh.v76i7.7536 |