Monogenic Cyclic Polynomials in Recurrence Sequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909610513793024 |
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| author | Harrington, Joshua Jones, Lenny |
| author_facet | Harrington, Joshua Jones, Lenny |
| contents | Let $f(x)\in {\mathbb Z}[x]$ be an $N$th degree polynomial that is monic and irreducible over ${\mathbb Q}$. We say that $f(x)$ is {\em monogenic} if $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. We say that $f(x)$ is {\em cyclic} if the Galois group of $f(x)$ over ${\mathbb Q}$ is the cyclic group of order $N$. In this article, we investigate the appearance of monogenic cyclic polynomials in certain polynomial recurrence sequences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_09481 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monogenic Cyclic Polynomials in Recurrence Sequences Harrington, Joshua Jones, Lenny Number Theory Let $f(x)\in {\mathbb Z}[x]$ be an $N$th degree polynomial that is monic and irreducible over ${\mathbb Q}$. We say that $f(x)$ is {\em monogenic} if $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. We say that $f(x)$ is {\em cyclic} if the Galois group of $f(x)$ over ${\mathbb Q}$ is the cyclic group of order $N$. In this article, we investigate the appearance of monogenic cyclic polynomials in certain polynomial recurrence sequences. |
| title | Monogenic Cyclic Polynomials in Recurrence Sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.09481 |