I want to understand one of the properties of the action of a group ring on cyclotomic Galois extension.
If $\zeta_n$ is primitive root of unity, $G = Gal(\mathbb{Q}(\zeta_n) / \mathbb{Q})$ and $f : G \rightarrow \mathbb{Z}$ is an element of the group ring $\mathbb{Z}[G]$, the action of $f$ on an element $x \in \mathbb{Q}(\zeta_n)$ is defined as: $$x^f = \prod_{\sigma \in G} \sigma(x)^{f(\sigma)}$$
The property I'm trying to prove is:
$$(x_1 + x_2)^f = x_1^f + x_2^f$$
My calculations go like this:
$$(x_1 + x_2)^f = \prod_{\sigma \in G}\sigma(x_1 + x_2)^{f(\sigma)} = \prod_{\sigma \in G}(\sigma(x_1) + \sigma(x_2))^{f(\sigma)}$$
$$x_1^f + x_2^f = \prod_{\sigma \in G} \sigma(x_1)^{f(\sigma)} + \prod_{\sigma \in G} \sigma(x_2)^{f(\sigma)}$$
Don't know how to finish the proof. Any help with this? Thanks in advance!