On the discriminant and index of a certain class of polynomials
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918342991806464 |
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| author | Barman, Rupam Narode, Anuj Wagh, Vinay |
| author_facet | Barman, Rupam Narode, Anuj Wagh, Vinay |
| contents | Let $f(x) = (x^{2}+1)^{n} - a x^{n} \in \mathbb{Z}[x]$ and assume $f(x)$ is irreducible. Let $θ$ be a root of $f(x)$, set $K= \mathbb{Q}(θ)$, and denote by $\mathbb{Z}_{K}$ the ring of integers of $K$. The index of $f$, denoted $\operatorname{ind}(f)$, is the index of $\mathbb{Z}[θ]$ in $\mathbb{Z}_{K}$. A polynomial $f(x)$ is said to be monogenic if $\operatorname{ind}(f) = 1$. In this article, we explicitly compute the discriminant of the polynomial $f(x)$, and then derive necessary and sufficient conditions on the parameters $a$ and $n$ for $f(x)$ to be monogenic. Furthermore, we provide a complete description of the primes that divide $\operatorname{ind}(f)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15641 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the discriminant and index of a certain class of polynomials Barman, Rupam Narode, Anuj Wagh, Vinay Number Theory Let $f(x) = (x^{2}+1)^{n} - a x^{n} \in \mathbb{Z}[x]$ and assume $f(x)$ is irreducible. Let $θ$ be a root of $f(x)$, set $K= \mathbb{Q}(θ)$, and denote by $\mathbb{Z}_{K}$ the ring of integers of $K$. The index of $f$, denoted $\operatorname{ind}(f)$, is the index of $\mathbb{Z}[θ]$ in $\mathbb{Z}_{K}$. A polynomial $f(x)$ is said to be monogenic if $\operatorname{ind}(f) = 1$. In this article, we explicitly compute the discriminant of the polynomial $f(x)$, and then derive necessary and sufficient conditions on the parameters $a$ and $n$ for $f(x)$ to be monogenic. Furthermore, we provide a complete description of the primes that divide $\operatorname{ind}(f)$. |
| title | On the discriminant and index of a certain class of polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.15641 |