Is there any isomorphism between the quotient ring $F{_p}$ /$\left\langle {{x^n} - a}\right\rangle$ and the group algebra $F{_p}G$

Question

We know that the quotient ring $F{_p}$ /$\left\langle {{x^n} - 1}\right\rangle$ is isomorphic to the group algebra $F{_p}C{_n},$ where $F{_p}$ is a finite field of characteristic $p$ and $C{_n}$ is a cyclic group of order $n.$ My question is that: Can we construct a group algebra $F{_p}G,$ which is isomorphic to the quotient ring $F{_p}$ /$\left\langle {{x^n} - a}\right\rangle$ where $a\in F{_p}$ and $a\neq 0,1.$

Answer 1

Let $\mathbb{Z}_{10}=\left\{ 0,1,2,3,4,5,6,7,8,9\right\} $ be the set of integers modulo $10$ and $G=2\mathbb{Z}_{10}^{\ast}=\left\{ 2,4,6,8\right\} \subset \mathbb{Z}_{10}$ be the set of all doubled elements in $\mathbb{Z}_{10}^{\ast }.$ The set $G$ is a cyclic multiplicative group with identity element $6.$ Thus, there exists an element $g\in G$ such that $g^{4}=e$ and $g^{i}\neq e$ for $i=1,2,3.$ Let $% \mathbb{F}_{7}$ be the finite field of characteristic $7.$ Then the group algebra $\mathbb{F}_{7}G$ is the set of all elements of the form $% \sum\limits_{i=0}^{3}\alpha _{i}g^{i},$ where $\alpha _{i}\in \mathbb{F}_{7}$ for $i=0,1,2,3.$ And so, we have the isomorphism $\mathbb{F}_{7}[x]/\left\langle {x}^{4}{-6}\right\rangle \cong \mathbb{F}_{7}G.$