I have the following problem:
Let $R$ be a commutative ring with identity and $\phi: R \rightarrow R$ a ring automorphism. If $F=\lbrace r\in R | \phi(r)=r \rbrace$, show that $\phi^2$ being the identity map implies each element of $R$ is the root of a monic polynomial of degree two in $F[x]$.
I've tried constructing the polynomial explicitly, but I haven't had any luck. I've considered trying to use an argument using indices, but I'm not sure how to go about it.