A generalization of Ore's theorem on polynomials

A. V. Anashkin

Diskretnaya Matematika, Tome 27 (2015) no. 4, pp. 21-25

Résumé

Let $GF(q)$ be the field of $q$ elements and ${V_n}(q)$ denote the $n$-dimensional vector space over the field $GF(q)$. The linearized polynomial that corresponds to the polynomial $f(x) = {x^n} - \sum\limits_{i = 0}^{n - 1} {{c_i}{x^i}} \;$over the field $GF(q)$ is the polynomial $F(x) = {x^{{q^n}}} - \sum\limits_{i = 0}^{n - 1} {{c_i}{x^{{q^i}}}}$. Let ${T_f}$ denote the transformation of the vector space ${V_n}(q)$ determined by the rule ${T_f}\left( {({u_0},...,{u_{n - 2}},{u_{n - 1}})} \right) = ({u_1},...,{u_{n - 1}},\sum\limits_{i = 0}^{n - 1} {{c_i}{u_i}} )$. It is shown that if ${c_0} \ne 0$, then the graph of the transformation ${T_f}$ is isomorphic to the graph of the transformation $Q:\alpha \to {\alpha ^q}$ on the set of all roots of the polynomial $F(x)$ in its splitting field. In this case the graph of the transformation ${T_f}$ consists of cycles of lengths $1 \le {d_1} \le {d_2} \le ... \le {d_r}$ if and only if the polynomial $F(x)$ is the product of $r + 1$ irreducible factors of degrees $1,{d_1},{d_2},...,{d_r}$.

Détail

Keywords: linearized polynomial, primitive polynomial, isomorphism of graphs, Ore's theorem.