Monogenic sextic trinomials $x^6+Ax^3+B$ and their Galois groups
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915405593837568 |
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| author | Harrington, Joshua Jones, Lenny |
| author_facet | Harrington, Joshua Jones, Lenny |
| contents | Let $f(x)=x^6+Ax^3+B\in {\mathbb Z}[x]$, with $A\ne 0$, and suppose that $f(x)$ is irreducible over ${\mathbb Q}$. We define $f(x)$ to be {\em monogenic} if $\{1,θ,θ^2,θ^3,θ^4,θ^{5}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$.
For each possible Galois group $G$ of $f(x)$ over ${\mathbb Q}$, we use a theorem of Jakhar, Khanduja and Sangwan to give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. We also investigate when these trinomials generate distinct sextic fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17021 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monogenic sextic trinomials $x^6+Ax^3+B$ and their Galois groups Harrington, Joshua Jones, Lenny Number Theory Let $f(x)=x^6+Ax^3+B\in {\mathbb Z}[x]$, with $A\ne 0$, and suppose that $f(x)$ is irreducible over ${\mathbb Q}$. We define $f(x)$ to be {\em monogenic} if $\{1,θ,θ^2,θ^3,θ^4,θ^{5}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. For each possible Galois group $G$ of $f(x)$ over ${\mathbb Q}$, we use a theorem of Jakhar, Khanduja and Sangwan to give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. We also investigate when these trinomials generate distinct sextic fields. |
| title | Monogenic sextic trinomials $x^6+Ax^3+B$ and their Galois groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.17021 |