Automorphisms of a ring

#ring_theory

Let $R$ be a commutative ring with $1$ , and $σ : R \to R$ be a ring automorphism.
(a) Show that $F = {r \in R ∣ σ (r) = r}$ is a subring of $R$ (with $1$ ).
(b) Show that if $σ^{2}$ is the identity map on $R$ , then each element of $R$ is the root of a monic polynomial of degree 2 in $F [x]$ , where $F$ is as in (a).

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Let $R$ be a commutative ring with $1$, and $\sigma:R\to R$ be a ring automorphism.
\begin{enumerate}[label=\alph*)]
	\item Show that $F=\{r\in R\mid \sigma(r)=r\}$ is a subring of $R$ (with $1$).
	\item Show that if $\sigma^2$ is the identity map on $R$, then each element of $R$ is the root of a monic polynomial of degree 2 in $F[x]$, where $F$ is as in (a).
\end{enumerate}