Monogenic trinomials of the form $x^4+ax^3+d$ and their Galois groups
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| Format: | Preprint |
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2024
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| _version_ | 1866915011452993536 |
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| author | Harrington, Joshua Jones, Lenny |
| author_facet | Harrington, Joshua Jones, Lenny |
| contents | Let $f(x)=x^4+ax^3+d\in {\mathbb Z}[x]$, where $ad\ne 0$. Let $C_n$ denote the cyclic group of order $n$, $D_4$ the dihedral group of order 8, and $A_4$ the alternating group of order 12. Assuming that $f(x)$ is monogenic, we give necessary and sufficient conditions involving only $a$ and $d$ to determine the Galois group $G$ of $f(x)$ over ${\mathbb Q}$. In particular, we show that $G=D_4$ if and only if $(a,d)=(\pm 2,2)$, and that $G\not \in \{C_4,C_2\times C_2\}$. Furthermore, we prove that $f(x)$ is monogenic with $G=A_4$ if and only if $a=4k$ and $d=27k^4+1$, where $k\ne 0$ is an integer such that $27k^4+1$ is squarefree. This article extends previous work of the authors on the monogenicity of quartic polynomials and their Galois groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_00413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monogenic trinomials of the form $x^4+ax^3+d$ and their Galois groups Harrington, Joshua Jones, Lenny Number Theory Let $f(x)=x^4+ax^3+d\in {\mathbb Z}[x]$, where $ad\ne 0$. Let $C_n$ denote the cyclic group of order $n$, $D_4$ the dihedral group of order 8, and $A_4$ the alternating group of order 12. Assuming that $f(x)$ is monogenic, we give necessary and sufficient conditions involving only $a$ and $d$ to determine the Galois group $G$ of $f(x)$ over ${\mathbb Q}$. In particular, we show that $G=D_4$ if and only if $(a,d)=(\pm 2,2)$, and that $G\not \in \{C_4,C_2\times C_2\}$. Furthermore, we prove that $f(x)$ is monogenic with $G=A_4$ if and only if $a=4k$ and $d=27k^4+1$, where $k\ne 0$ is an integer such that $27k^4+1$ is squarefree. This article extends previous work of the authors on the monogenicity of quartic polynomials and their Galois groups. |
| title | Monogenic trinomials of the form $x^4+ax^3+d$ and their Galois groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.00413 |