Elementary characterization for Galois groups of $x^{12}+ax^6+b$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918116880023552 |
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| author | Chen, Malcolm Hoong Wai |
| author_facet | Chen, Malcolm Hoong Wai |
| contents | Let $f(x)=x^{12}+ax^6+b \in \mathbb{Q}[x]$ be an irreducible polynomial, $g_4(x)=x^4+ax^2+b$, $g_6(x)=x^6+ax^3+b$, and let $G_4$ and $G_6$ be the Galois group of $g_4(x)$ and $g_6(x)$, respectively. Building upon known characterizations of $G_4$ and $G_6$ in the literature, this paper provides an elementary characterization of all sixteen possible Galois groups of $f(x)$. In particular, we show that the Galois group of $f(x)$ can be uniquely determined by the pair $(G_4,G_6)$ along with testing whether at most two expressions involving $a$ and $b$ are rational squares. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elementary characterization for Galois groups of $x^{12}+ax^6+b$ Chen, Malcolm Hoong Wai Number Theory 12F10, 11R09, 12D05, 12-08 Let $f(x)=x^{12}+ax^6+b \in \mathbb{Q}[x]$ be an irreducible polynomial, $g_4(x)=x^4+ax^2+b$, $g_6(x)=x^6+ax^3+b$, and let $G_4$ and $G_6$ be the Galois group of $g_4(x)$ and $g_6(x)$, respectively. Building upon known characterizations of $G_4$ and $G_6$ in the literature, this paper provides an elementary characterization of all sixteen possible Galois groups of $f(x)$. In particular, we show that the Galois group of $f(x)$ can be uniquely determined by the pair $(G_4,G_6)$ along with testing whether at most two expressions involving $a$ and $b$ are rational squares. |
| title | Elementary characterization for Galois groups of $x^{12}+ax^6+b$ |
| topic | Number Theory 12F10, 11R09, 12D05, 12-08 |
| url | https://arxiv.org/abs/2410.00870 |