Action of Galois group permutes prime ideals

Question

Let $K/\mathbb{Q}$ be a Galois extension with ring of integers $R$ and Galois group $G=\text{Gal}(E/\mathbb{Q})$. How does one easily see that the action of $G$ on $E$ preserves $R$ and permutes the prime ideals of $R$?

Answer 1

Take $\sigma \in Gal(E/\mathbb Q)$. If $\alpha \in R$ is a root of a monic polynomial $P \in \mathbb Z[X]$, then $\sigma(\alpha)$ is a root of $\sigma(P) = P$.

Thus $\sigma$ restricts to $R$, and the restriction is an automorphism of $R$ (with inverse $\sigma^{-1}|_R$). Isomorphisms between rings send prime ideals to prime ideals.