On the index of power compositional polynomials
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912368205758464 |
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| author | Kaur, Sumandeep Kumar, Surender Remete, László |
| author_facet | Kaur, Sumandeep Kumar, Surender Remete, László |
| contents | The index of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ having a root $θ$ is the index $[\mathbb{Z}_K:\mathbb{Z}[θ]]$, where $\mathbb{Z}_K$ is the ring of algebraic integers of the number field $K=\mathbb{Q}(θ)$. If $[\mathbb{Z}_K:\mathbb{Z}[θ]]=1$, then $f(x)$ is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial $f(x^k)$ belonging to $\mathbb{Z}[x]$, to be monogenic. As an application of our results, for a polynomial $f(x)=x^d+A\cdot h(x)\in\mathbb{Z}[x],$ with $d>1, \operatorname{deg} h(x)<d$ and $|h(0)|=1$, we prove that for each positive integer $k$ with $\operatorname{rad}(k)\mid \operatorname{rad}(A)$, the power compositional polynomial $f(x^k)$ is monogenic if and only if $f(x)$ is monogenic, provided that $f(x^k)$ is irreducible. At the end of the paper, we give infinite families of polynomials as examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the index of power compositional polynomials Kaur, Sumandeep Kumar, Surender Remete, László Number Theory Commutative Algebra 11R04, 11R09 The index of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ having a root $θ$ is the index $[\mathbb{Z}_K:\mathbb{Z}[θ]]$, where $\mathbb{Z}_K$ is the ring of algebraic integers of the number field $K=\mathbb{Q}(θ)$. If $[\mathbb{Z}_K:\mathbb{Z}[θ]]=1$, then $f(x)$ is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial $f(x^k)$ belonging to $\mathbb{Z}[x]$, to be monogenic. As an application of our results, for a polynomial $f(x)=x^d+A\cdot h(x)\in\mathbb{Z}[x],$ with $d>1, \operatorname{deg} h(x)<d$ and $|h(0)|=1$, we prove that for each positive integer $k$ with $\operatorname{rad}(k)\mid \operatorname{rad}(A)$, the power compositional polynomial $f(x^k)$ is monogenic if and only if $f(x)$ is monogenic, provided that $f(x^k)$ is irreducible. At the end of the paper, we give infinite families of polynomials as examples. |
| title | On the index of power compositional polynomials |
| topic | Number Theory Commutative Algebra 11R04, 11R09 |
| url | https://arxiv.org/abs/2404.17351 |