Monogenic cyclic trinomials of the form $x^4+cx+d$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909392617603072 |
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| author | Jones, Lenny |
| author_facet | Jones, Lenny |
| contents | A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ that is irreducible over ${\mathbb Q}$ is called cyclic if the Galois group over ${\mathbb Q}$ of $f(x)$ is the cyclic group of order $n$, while $f(x)$ is called monogenic if $\{1,θ,θ^2,\ldots, θ^{n-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. In this article, we show that there do not exist any monogenic cyclic trinomials of the form $f(x)=x^4+cx+d$. This result, combined with previous work, proves that the only monogenic cyclic quartic trinomials are $x^4-4x^2+2$, $x^4+4x^2+2$ and $x^4-5x^2+5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10572 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monogenic cyclic trinomials of the form $x^4+cx+d$ Jones, Lenny Number Theory A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ that is irreducible over ${\mathbb Q}$ is called cyclic if the Galois group over ${\mathbb Q}$ of $f(x)$ is the cyclic group of order $n$, while $f(x)$ is called monogenic if $\{1,θ,θ^2,\ldots, θ^{n-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. In this article, we show that there do not exist any monogenic cyclic trinomials of the form $f(x)=x^4+cx+d$. This result, combined with previous work, proves that the only monogenic cyclic quartic trinomials are $x^4-4x^2+2$, $x^4+4x^2+2$ and $x^4-5x^2+5$. |
| title | Monogenic cyclic trinomials of the form $x^4+cx+d$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.10572 |