Monogenic Fields from Polynomial Compositions with Applications
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911639710728192 |
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| author | Jakhar, Anuj Kalwaniya, Ravi Yadav, Prabhakar |
| author_facet | Jakhar, Anuj Kalwaniya, Ravi Yadav, Prabhakar |
| contents | A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[α]$ for some $α\in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for $[\mathbb{Z}_{K_i}:\mathbb{Z}[α_i]]=1$, where $K_i=\mathbb{Q}(α_i)$ and $α_i$ is a root of the composed polynomial $f_i(x^k+b)$ for $i=1,2$. Here, $f_1(x)=x^n+c\sum_{j=1}^{n}(ax)^{n-j}\in\mathbb{Z}[x]$ and $f_2(x)=x^n+c\sum_{j=1}^{n}a^{j-1}x^{n-j}\in\mathbb{Z}[x]$ are irreducible polynomials of degree $n\ge 3$. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Monogenic Fields from Polynomial Compositions with Applications Jakhar, Anuj Kalwaniya, Ravi Yadav, Prabhakar Number Theory 11R04, 11R29, 11Y40 A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[α]$ for some $α\in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for $[\mathbb{Z}_{K_i}:\mathbb{Z}[α_i]]=1$, where $K_i=\mathbb{Q}(α_i)$ and $α_i$ is a root of the composed polynomial $f_i(x^k+b)$ for $i=1,2$. Here, $f_1(x)=x^n+c\sum_{j=1}^{n}(ax)^{n-j}\in\mathbb{Z}[x]$ and $f_2(x)=x^n+c\sum_{j=1}^{n}a^{j-1}x^{n-j}\in\mathbb{Z}[x]$ are irreducible polynomials of degree $n\ge 3$. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations. |
| title | Monogenic Fields from Polynomial Compositions with Applications |
| topic | Number Theory 11R04, 11R29, 11Y40 |
| url | https://arxiv.org/abs/2605.00949 |