Why is $KG \cong K[x]/(x^p-1)$, for $K=\mathbb{Z}/p \mathbb{Z}$ and $G = \langle g | g^p=1 \rangle$ a ring isomorphism?
If I take $g \mapsto [x]$, I'd have an group homomorphism. And than an algebra(?) homomorphism f: $KG \rightarrow K[x]$, $a_g g \mapsto a_g[x]$, with $a_g \in K$. Now I need $\ker(f)=(x^p-1)$, not?