Let me start by saying I know that there are similar problems on here but none that give any real sense of direction or understanding, at least for me.
Let $F$ be a field such that $\text{char} (F)=p > 0$ and $G$ a finite cyclic group of order $n$.
Show that $F[G]$ is isomorphic to $F[x]/(x^n - 1)$ as an $F$-algebra, and find the Jacobson radical of $F[G]$.
I believe that I've shown that these are isomorphic as $F$ algebras. I however feel like I don't have any idea how this then helps me find the Jacobson radical. I know that it is the intersection of maximal left ideals but don't know if this is the right way to go about it or if it is how to find those ideals.