The monogenicity and Galois groups of certain reciprocal quintinomials
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909374576852992 |
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| author | Jones, Lenny |
| author_facet | Jones, Lenny |
| contents | We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for ${\mathbb Z}_K$, the ring of integers of $K={\mathbb Q}(θ)$, where $f(θ)=0$. For $n\ge 2$, we define the reciprocal quintinomial \[{\mathcal F}_{n,A,B}(x):=x^{2^n}+Ax^{3\cdot 2^{n-2}}+Bx^{2^{n-1}}+Ax^{2^{n-2}}+1\in {\mathbb Z}[x].\] In this article, we extend our previous work on the monogenicity of ${\mathcal F}_{n,A,B}(x)$ to treat the specific previously-unaddressed situation of $A\equiv B\equiv 1\pmod{4}$. Moreover, we determine the Galois group over ${\mathbb Q}$ of ${\mathcal F}_{n,A,B}(x)$ in special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00523 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The monogenicity and Galois groups of certain reciprocal quintinomials Jones, Lenny Number Theory We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for ${\mathbb Z}_K$, the ring of integers of $K={\mathbb Q}(θ)$, where $f(θ)=0$. For $n\ge 2$, we define the reciprocal quintinomial \[{\mathcal F}_{n,A,B}(x):=x^{2^n}+Ax^{3\cdot 2^{n-2}}+Bx^{2^{n-1}}+Ax^{2^{n-2}}+1\in {\mathbb Z}[x].\] In this article, we extend our previous work on the monogenicity of ${\mathcal F}_{n,A,B}(x)$ to treat the specific previously-unaddressed situation of $A\equiv B\equiv 1\pmod{4}$. Moreover, we determine the Galois group over ${\mathbb Q}$ of ${\mathcal F}_{n,A,B}(x)$ in special cases. |
| title | The monogenicity and Galois groups of certain reciprocal quintinomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.00523 |