Q:

Prove that if $f\in R[X]$ , then $\displaystyle\prod _{\sigma \in G}f^{\sigma}\in \mathbb{Z}[X].$

Let $K$ be an algebraic number field and $R$ be the ring of algebraic integers of $K.$ Denote by $h^{\sigma}$ the polynomial obtained from $h\in  K[X]$ after applying to its coefficients the $\mathbb{Q}$-automorphism $\sigma \in G.$ Prove that if $f\in R[X]$ , then $\displaystyle\prod _{\sigma \in G}f^{\sigma}\in \mathbb{Z}[X].$

A:

Hint: set $g=\prod_{\sigma\in G}f^{\sigma}$. Now fix $\tau \in G$. What is $g^{\tau}$?

