Galois groups of certain even octic polynomials
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912927059017728 |
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| author | Chen, Malcolm Hoong Wai Chin, Angelina Yan Mui Tan, Ta Sheng |
| author_facet | Chen, Malcolm Hoong Wai Chin, Angelina Yan Mui Tan, Ta Sheng |
| contents | Let $f(x)=x^8+ax^4+b \in \mathbb{Q}[x]$ be an irreducible polynomial where $b$ is a square. We give a method that completely describes the factorization patterns of a linear resolvent of $f(x)$ using simple arithmetic conditions on $a$ and $b$. As a result, we determine the exact six possible Galois groups of $f(x)$ and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials $x^8+ax^4+1 \in \mathbb{Q}[x]$. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials $x^8+ax^6+bx^4+ax^2+1 \in \mathbb{Q}[x]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_10257 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Galois groups of certain even octic polynomials Chen, Malcolm Hoong Wai Chin, Angelina Yan Mui Tan, Ta Sheng Number Theory 12F10, 11R09, 12D05, 12-08 Let $f(x)=x^8+ax^4+b \in \mathbb{Q}[x]$ be an irreducible polynomial where $b$ is a square. We give a method that completely describes the factorization patterns of a linear resolvent of $f(x)$ using simple arithmetic conditions on $a$ and $b$. As a result, we determine the exact six possible Galois groups of $f(x)$ and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials $x^8+ax^4+1 \in \mathbb{Q}[x]$. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials $x^8+ax^6+bx^4+ax^2+1 \in \mathbb{Q}[x]$. |
| title | Galois groups of certain even octic polynomials |
| topic | Number Theory 12F10, 11R09, 12D05, 12-08 |
| url | https://arxiv.org/abs/2210.10257 |