Dual And Non-Dual Elemeents In Finite Fields (Rings)

Abstract

Let $F$ be a finite field (ring) and $a,b\in F$. We call $a$ and $b$ as dual elements if $a^{2} =b^{2} =-1$ ( where $1$ is the identity element of $F$). The term dual elements refers to the dual properties of $a$ and $b$ as $a$ and $b$ are the additive as well as multiplicative inverse of each other. If $a^{2} =b^{2} =-c$, where $c$ is any element of $F$ then we call $a$ and $b$ as non-dual elements of $F$. We note that if $a\in F$ such that $a^{2} =-a$ then $a$ is not necessarily the zero element of $F$.