Let $F$ be a field. I'd like to show that:
$a)$ If $V$ is a cyclic group of order $n$, then $$F[V] \cong F[x]/(x^n - 1).$$ $b)$ If $\operatorname{char}(F)\neq 2$, $V=\{1, f, g, fg\}$ is the Klein group, and $\phi : F[x,y] \to F[V]$ is a morphism such that $\phi(x)=1+f$ and $\phi(y)=1+g$; then $$F[V] \cong F[x,y]/(x^2,y^2).$$
My attempt : for $a)$ if $\varepsilon:F[x]\to F[V]$ is a morphism, then $\ker(\varepsilon)$ is an ideal of $F[x]$. Since $F[x]$ is a PID, then there exists $m \in F[x]$ monic such that $\ker(\varepsilon)=(m)$. For $b)$ I think that the kernel of of $\phi$ is $(x^2,y^2)$: we have $\phi(x^2)=\phi(y^2)=0$.
Thanks for any help!