On cubic polynomials with the cyclic Galois group

Yury Kochetkov

Published 2024-01-20Version 1

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.