Let $K$ be a number field and $\Delta = Gal (K/\mathbb{Q})$ and $\chi: \Delta \rightarrow \mathbb{Z}_p^*$ be a non-trivial Dirichlet character, $e_{\chi} = (1/\mid \Delta \mid) \sum_{\sigma \in \Delta} \chi (\sigma) \sigma$ be the corresponding idempotent.

I am trying to understand why the following statement is true:

Let $p$ be a prime such that $p \not \mid [K:\mathbb{Q}]$. Since $K \subseteq \mathbb{Q}(\zeta_p) \cap \mathbb{R}$, we have that $\sum_{\chi} e_{\chi} = 1$, where $\chi$ runs over all $p$-adic Dirichlet characters of $\Delta$.

PS it's on page 15 of the paper of Thaine on Ideal Class Groups of real abelian number fields in Annals of Math, 128.