I'm stuck on the proof of Theorem 4.14 in Washington - Introduction to Cyclotomic Fields.
We take a prime power $n = p^m$ and define $\pi = \zeta_n - 1$, $\zeta_n$ a primitive $n$-th root of unity. Let $\mathbb{Q}(\zeta_n)^+ = \mathbb{Q}(\zeta_n + \zeta_n^{-1})$ be the maximal real subfield. The proof claims that the $\pi$-adic valuation of $\mathbb{Q}(\zeta_n)$ only takes on even values on $\mathbb{Q}(\zeta_n)^+$. I'm not sure how to show this.
Thanks for any help.