On cubic polynomials with the cyclic Galois group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929218034597888 |
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| author | Kochetkov, Yury |
| author_facet | Kochetkov, Yury |
| contents | A cubic Galois polynomial is a cubic polynomial with rational coefficients that defines a cubic Galois field. Its discriminant is a full square and its roots $x_1,x_2,x_3$ (enumerated in some order) are real. There exists (and only one) quadratic polynomial $q$ with rational coefficients such that $q(x_1)=x_2, q(x_2)=x_3, q(x_3)=x_1$. The polynomial $r=q(q)\text{ mod } p$ cyclically permutes roots of $p$ in the opposite order: $r(x_1)=x_3, r(x_3)=x_2, r(x_2)=x_1$. We prove that there exist a unique Galois polynomial $p_1$ and a unique Galois polynomial $p_2$ such that the polynomial $q$ cyclically permutes roots of $p_1$ and the polynomial $r$ do the same with roots of $p_2$. Polynomials $p$ and $p_1$ (and also $p$ and $p_2$) will be called \emph{coupled}. Two polynomials are \emph{linear equivalent}, if one of them is obtained from another by a linear change of variable. By $C(p)$ we denote the class of polynomials, linear equivalent to $p$. The coupling realizes a bijection between classes $C(p)$ and $C(p_1)$ (and between classes $C(p)$ and $C(p_2)$). Classes $C(p)$ and $C(p_1)$ (and classes $C(p)$ and $C(p_2)$) will be called \emph{adjacent}. We consider a graph: its vertices -- are classes of the linear equivalency and two vertices are connected by an edge, if the corresponded classes are adjacent. Connected components of this graph will be called \emph{superclasses}. In this work we give a description of superclasses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11208 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On cubic polynomials with the cyclic Galois group Kochetkov, Yury Number Theory A cubic Galois polynomial is a cubic polynomial with rational coefficients that defines a cubic Galois field. Its discriminant is a full square and its roots $x_1,x_2,x_3$ (enumerated in some order) are real. There exists (and only one) quadratic polynomial $q$ with rational coefficients such that $q(x_1)=x_2, q(x_2)=x_3, q(x_3)=x_1$. The polynomial $r=q(q)\text{ mod } p$ cyclically permutes roots of $p$ in the opposite order: $r(x_1)=x_3, r(x_3)=x_2, r(x_2)=x_1$. We prove that there exist a unique Galois polynomial $p_1$ and a unique Galois polynomial $p_2$ such that the polynomial $q$ cyclically permutes roots of $p_1$ and the polynomial $r$ do the same with roots of $p_2$. Polynomials $p$ and $p_1$ (and also $p$ and $p_2$) will be called \emph{coupled}. Two polynomials are \emph{linear equivalent}, if one of them is obtained from another by a linear change of variable. By $C(p)$ we denote the class of polynomials, linear equivalent to $p$. The coupling realizes a bijection between classes $C(p)$ and $C(p_1)$ (and between classes $C(p)$ and $C(p_2)$). Classes $C(p)$ and $C(p_1)$ (and classes $C(p)$ and $C(p_2)$) will be called \emph{adjacent}. We consider a graph: its vertices -- are classes of the linear equivalency and two vertices are connected by an edge, if the corresponded classes are adjacent. Connected components of this graph will be called \emph{superclasses}. In this work we give a description of superclasses. |
| title | On cubic polynomials with the cyclic Galois group |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.11208 |