The Cocycle structure of the Alexander $f$-quandles on finite fields

We determine the second, third, and fourth cohomology groups of Alexander $f$-quandles of the form ${\mathbb{F}}_q[T,S]\mathrm{/}(T-\omega ,S-\beta )$, where ${\mathbb{F}}_q$ denotes the finite field of order $q$, $\omega \in $\mathbb{F}$_q\setminus {0,1}$,and$\beta \in $\mathbb{F}$_q$