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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2401.13925 |
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| _version_ | 1866917574405521408 |
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| author | Jim, Akash Hagedorn, Thomas |
| author_facet | Jim, Akash Hagedorn, Thomas |
| contents | We prove an irreducibility criterion for polynomials of the form $h(x)=x^{2m} + bx^m + c_1 \in F[x]$ relating to the Dickson polynomials of the first kind $D_p$. In the case when $F = \mathbb{Q}$, $m$ is a prime $p>3$, and $c_1=c^p$, for $c\in\mathbb{Q}$, we explicitly determine the Galois group of $d_h= D_p(x, c) + b$, which is $\mathrm{Aff}(\mathbb{F}_p)$ or $C_p \rtimes C_{(p - 1)/2} \vartriangleleft \mathrm{Aff}(\mathbb{F}_p)$, and the Galois group of $h$, which is $C_2 \times \mathrm{Aff}(\mathbb{F}_p), \mathrm{Aff}(\mathbb{F}_p)$, or $C_2 \times (C_p \rtimes C_{(p - 1)/2}) \vartriangleleft C_2 \times \mathrm{Aff}(\mathbb{F}_p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13925 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Galois Group of $x^{2p}+bx^p+c^p$ over $\mathbb{Q}$ Jim, Akash Hagedorn, Thomas Number Theory 12F10, 12E05, 11R09 We prove an irreducibility criterion for polynomials of the form $h(x)=x^{2m} + bx^m + c_1 \in F[x]$ relating to the Dickson polynomials of the first kind $D_p$. In the case when $F = \mathbb{Q}$, $m$ is a prime $p>3$, and $c_1=c^p$, for $c\in\mathbb{Q}$, we explicitly determine the Galois group of $d_h= D_p(x, c) + b$, which is $\mathrm{Aff}(\mathbb{F}_p)$ or $C_p \rtimes C_{(p - 1)/2} \vartriangleleft \mathrm{Aff}(\mathbb{F}_p)$, and the Galois group of $h$, which is $C_2 \times \mathrm{Aff}(\mathbb{F}_p), \mathrm{Aff}(\mathbb{F}_p)$, or $C_2 \times (C_p \rtimes C_{(p - 1)/2}) \vartriangleleft C_2 \times \mathrm{Aff}(\mathbb{F}_p)$. |
| title | The Galois Group of $x^{2p}+bx^p+c^p$ over $\mathbb{Q}$ |
| topic | Number Theory 12F10, 12E05, 11R09 |
| url | https://arxiv.org/abs/2401.13925 |