In the proof of Theorem 7.2 in the Appendix by Rubin on the Iwasawa Main Conjecture in Lang's Cyclotomic Fields, it is claimed that
For an even nontrivial character $\chi$ of $\Delta$, $\xi^{e(\chi)}$ generates $e(\chi)V_n$ where
- for a character $\chi$ of $\Delta=\mathrm{Gal}(\mathbb Q(\zeta_p)/\mathbb Q)\simeq(\mathbb Z/p\mathbb Z)^\times$ where $\zeta_p$ is a primitive $p$th root of unity, $e(\chi)=\frac 1 {p-1}\sum_{\delta\in\Delta} \chi^{-1}(\delta)\delta$ is an orthogonal idempotent,
- $\xi=(\zeta_{p^n}-1)(\zeta^{-1}_{p^n}-1)$ where $\zeta_{p^n}$ is a primitive ${p^n}$th root of unity,
- $V_n$ is the closure of $\mathcal E_n \cap U_n$ in $U_n$ where $E_n$ is the group of cyclotomic units of $\mathbb Q(\zeta_{p^n+1})$ and $U_n$ is the group of local units in the completion of $\mathbb Q(\zeta_{p^n+1})$ above $p$ congruent to $1$ modulo the maximal ideal
The same statement is made in Washington's Introduction to Cyclotomic Fields on page 370. Neither book gives any further explanation as to why this holds. Could someone please help me understand?