Monogenic Reciprocal Quartic Polynomials And Their Galois Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909509482446848 |
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| author | Jones, Lenny |
| author_facet | Jones, Lenny |
| contents | Suppose that $f(x)=x^4+Ax^3+Bx^2+Ax+1\in {\mathbb Z}[x]$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,θ^3\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. For each possible Galois group $G$ that can occur in the two cases of $A\ne 0$ with $B=0$, and $AB\ne 0$, we determine all monogenic polynomials $f(x)$ with Galois group $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17691 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monogenic Reciprocal Quartic Polynomials And Their Galois Groups Jones, Lenny Number Theory Suppose that $f(x)=x^4+Ax^3+Bx^2+Ax+1\in {\mathbb Z}[x]$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,θ^3\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. For each possible Galois group $G$ that can occur in the two cases of $A\ne 0$ with $B=0$, and $AB\ne 0$, we determine all monogenic polynomials $f(x)$ with Galois group $G$. |
| title | Monogenic Reciprocal Quartic Polynomials And Their Galois Groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.17691 |