1
$\begingroup$

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 $a\neq 0,1.$

$\endgroup$
0
$\begingroup$

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 $\mathbb{F}_{7}[x]/\left\langle {x}^{4}{-6}\right\rangle \cong \mathbb{F}_{7}G.$ For further information see http://www.pmf.ni.ac.rs/pmf/publikacije/filomat/2017/31-10/31-10-2.pdf

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.