Group ring of a cyclic group over a finite field

Question

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?

Answer 1

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.