A Study of monogenity of Binomial Composition
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911777723252736 |
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| author | Jakhar, Anuj Kalwaniya, Ravi Yadav, Prabhakar |
| author_facet | Jakhar, Anuj Kalwaniya, Ravi Yadav, Prabhakar |
| contents | Let $θ$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n \geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1, θ, θ^2, \ldots, θ^{n-1} \}$ is a basis for the ring $\Z_K$ of integers of $K = \Q(θ)$. In this article, we study about the monogenity of number fields generated by a root of composition of two binomials. We characterise all the primes dividing the index of the subgroup $\Z[θ]$ in $\Z_K$ where $K = \Q(θ)$ with $θ$ having minimal polynomial $F(x) = (x^m-b)^n - a \in \Z[x]$, $m\geq 1$ and $n \geq 2$. As an application, we provide a class of pairs of binomials $f(x)=x^n-a$ and $g(x)=x^m-b$ having the property that both $f(x)$ and $f(g(x))$ are monogenic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10131 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Study of monogenity of Binomial Composition Jakhar, Anuj Kalwaniya, Ravi Yadav, Prabhakar Number Theory 11R04, 11R29, 11Y40 Let $θ$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n \geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1, θ, θ^2, \ldots, θ^{n-1} \}$ is a basis for the ring $\Z_K$ of integers of $K = \Q(θ)$. In this article, we study about the monogenity of number fields generated by a root of composition of two binomials. We characterise all the primes dividing the index of the subgroup $\Z[θ]$ in $\Z_K$ where $K = \Q(θ)$ with $θ$ having minimal polynomial $F(x) = (x^m-b)^n - a \in \Z[x]$, $m\geq 1$ and $n \geq 2$. As an application, we provide a class of pairs of binomials $f(x)=x^n-a$ and $g(x)=x^m-b$ having the property that both $f(x)$ and $f(g(x))$ are monogenic. |
| title | A Study of monogenity of Binomial Composition |
| topic | Number Theory 11R04, 11R29, 11Y40 |
| url | https://arxiv.org/abs/2402.10131 |