The 3–cocycles of the Alexander quandles $\mathbb{F}_q[T]/(T-\omega)$

Abstract

We determine the third cohomology of Alexander quandles of the form ${\mathbb{F}}_q\left[T\right]∕\left(T-\omega \right)$, where ${\mathbb{F}}_q$ denotes the finite field of order $q$ and $\omega$ is an element of ${\mathbb{F}}_q$ which is neither $0$ nor $1$. As a result, we obtain many concrete examples of non-trivial $3$–cocycles.