11.3. Generators for the global cyclotomic units
Recall the global cyclotomic units \DD_n=\cO_{F_n}^\times\cap\ang{\pm\xi_{p^n},\
\xi_{p^n}^a-1} and \DD_n^+=\DD_n\cap F_n^+. Set
c_n(a):=\tfrac{\xi_{p^n}^a-1}{\xi_{p^n}-1}\in\DD_n and the conjugation-invariant
\gamma_{n,a} := \xi_{p^n}^{(1-a)/2}c_n(a)
= \frac{\xi_{p^n}^{a/2}-\xi_{p^n}^{-a/2}}{\xi_{p^n}^{1/2}-\xi_{p^n}^{-1/2}} \in \DD_n^+.
Let n\ge 1. Then: (i) \DD_n^+ is generated by -1 together with
\set{\gamma_{n,a} : 1<a<p^n/2,\ (a,p)=1}; and (ii) \DD_n is generated by
\xi_{p^n} and \DD_n^+.
First reduce to a prime to p: the identity
\xi_{p^n}^{bp^m}-1=\prod_{j=0}^{p^m-1}(\xi_{p^n}^{\,b+jp^{n-m}}-1) (with (b,p)=1)
expresses the p-divisible exponents in terms of prime-to-p ones, and
\xi_{p^n}^a-1=-\xi_{p^n}^a(\xi_{p^n}^{-a}-1) lets us take 1\le a<p^n/2. Now write a
general element as \gamma=\pm\xi_{p^n}^d\prod_a(\xi_{p^n}^a-1)^{e_a}. All the
\xi_{p^n}^a-1 have equal p-adic valuation \tfrac{1}{(p-1)p^{n-1}} while
\xi_{p^n}^d is a unit, so being in \DD_n forces \sum_a e_a=0. Hence we may
divide each factor by \xi_{p^n}-1 and rewrite
\gamma=\pm\xi_{p^n}^{e}\prod_a\gamma_{n,a}^{e_a} with
e=d+\tfrac12\sum_a e_a(a-1), proving (ii). Each \gamma_{n,a} is real, so
\gamma\in\DD_n^+ iff e=0, giving (i).
If a generates (\Z/p^n\Z)^\times, then \gamma_{n,a} generates \DD_n^+ as
a \Z[\GGam_n^+]-module.
Any b prime to p is b\equiv a^r \pmod{p^n} for some r, and the telescoping
product \gamma_{n,b}=\prod_{i=0}^{r-1}\tfrac{\xi_{p^n}^{a^{i+1}}-1}{\xi_{p^n}^{a^i}-1}
=\prod_{i=0}^{r-1}(\gamma_{n,a})^{\sigma_a^i} exhibits every generator from
Lemma 11.3.1 as a \Z[\GGam_n^+]-translate of \gamma_{n,a}.