# Group ring of a cyclic group over a finite field

Suppose $p$ a prime integer and $n$ a positive integer. Does anyone know off the top of their heads if the group ring $\mathbb{F}_{p}[\mathbb{Z}/n]$ (perhaps regarding $\mathbb{Z}/n$ as the $n^{\mathrm{th}}$ roots of unity is helpful) is isomorphic to $\mathbb{F}_{p^n}$ in general?

• This is not true. The splitting into summands depends on whether $p$ is a factor of $n$ and (more importantly) what is the smallest extension field of $\Bbb{F}_p$ that contains $n$th roots of unity (when $p$ and $n$ are coprime). Anyway, the block of the trivial representation will always be there, so the group ring cannot be a field. – Jyrki Lahtonen May 22 '15 at 5:18
• See this answer for an explanation of what happens when $\gcd(n,p)=1$. – Jyrki Lahtonen May 22 '15 at 5:19

For any field $F$ and a cyclic group G of order $n$ the group ring is the quotient of the polynomial ring in a varisble X by the ideal generated by $X^ n-1$. As this polynomial is reducible this quotient ring will have zero divisors and hence cannot be field.