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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?

Hermetically Sealed Halibut
  • 560
  • I don't get what is $V_n $ and "generates" in what sense, $\zeta_{p^n}-1$ is an uniformizer of $\Bbb{Z}p[\zeta{p^n}]$ which is totally ramified so that $\xi$ is an uniformizer of $\Bbb{Z}_p[\cos(2\pi / p^n)]$, trivial or non-trivial $\chi$ should be different – reuns Sep 16 '19 at 00:24
  • I edited the (originally rather sloppily posed) question, hopefully it is clearer now. – Hermetically Sealed Halibut Sep 16 '19 at 07:58
  • ... which chapter is it in Washington's book, I don't think $U_n=1+\pi_KO_K,K=\mathbb Q(\zeta_{p^n+1}) $ make sense – reuns Sep 16 '19 at 13:28
  • It is on page 370 in Chapter 15, Section 7, in the proof of the Iwasawa Main Conjecture. In his notation, the statement is that $\varepsilon_\chi \kappa(1)$ generates $\varepsilon_\chi \overline{C}_1^n$. – Hermetically Sealed Halibut Sep 16 '19 at 20:54
  • What is your level in p-adic extensions ? Do you understand when I say $\pi=\zeta_{p^n}-1$ is an uniformizer of $O_K=\Bbb{Z}p[\zeta{p^n}]$ where $K=\Bbb{Q}p(\zeta{p^n})$ is totally ramified extension of $\Bbb{Q}p$ of degree $(p-1)p^{n-1}$ and that (for $p$ odd) $ 1+p\Bbb{Z}_p=(1+p)^{\Bbb{Z}_p}$,$1+\pi O_K = \prod{j=1}^{(p-1)p^{n-1}} (1+\pi^j)^{\Bbb{Z}_p}$ ? – reuns Sep 16 '19 at 21:12

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