Seemingly different definitions of cyclotomic units In the literature I see several definitions of cyclotomic units which becomes  confusing for me. My question is whether all the definitions are same.
Definition 1. (Reference here) Let $K$ be a cyclic real abelian number field with degree, conductor and the Galois group $g, f, G $ respectively. Let $H$ be the subgroup of $(\mathbb{Z}/f\mathbb{Z})^\times$ corresponding to Gal$(\mathbb{Q}(\zeta_f)/K)$, and let $H^{+} \subset \mathbb{Z}$ be a system of representatives of $H/\{\pm\}.$ Then the group of cyclotomic units $C_K$ is defined as 
$$C_K=\langle -1,\eta^\tau|\tau \in G\rangle$$
where $\eta=\Theta^\Lambda$,
$$\Theta=\prod_{a \in H^+}(\zeta_{2f}^a-\zeta_{2f}^{-a}),\quad \Lambda=\prod_{l \mid g}(1-\sigma^{g/l}),$$
$\sigma$ is a fixed generator of $G$ and $l$ runs through the prime divisors of $g$.
Definition 2 (Reference here) Here $C_K$ is defined as 
$$C_K=\langle -1,N(\eta_a):(a,2f)=1 \rangle $$
Here $\eta_a=(\zeta_{2f}-\zeta_{2f}^{-1})/(\zeta_{2f}^a-\zeta_{2f}^{-a})$ and $N:\mathbb{Q}(\zeta_f +\zeta_f^{-1}) \rightarrow K$ is the norm map.
Also I like to ask whether these two above definitions are equivalent to the following:
$$C_K=\left\langle -1, N_{\mathbb{Q}(\zeta_f)/K}\left(\frac{\zeta_f^a-1}{\zeta_f-1}\right) \right\rangle$$
where $a$ runs through all elements of $\mathbb{Z}/f\mathbb{Z}$.
Thanks in advance for your help. 
 A: I could not access to your ref.1 , and in your ref.2, only to p.1. However I can guess where the problem nests. The cyclotomic units of the totally real subfield of $\mathbf Q(\zeta_p)$ were introduced by Kummer in his study of FLT for the prime $p$. Around the 1970's,the generalization of this notion to totally real abelian (*) number fields became a priority, not in view of FLT (although in the end...), but for the needs of Iwasawa theory. The point is that the cyclo. units are more or less "explicit", and their index  inside the units is "equal", up to some parasite factors, to the class number . Many ad hoc definitions, according to what was needed, have been proposed, but in the end only two seem to have survived, those of Washington and of Sinnott (around the 80's). Although Sinnott came later than Washington, I begin by his circular units for clarity.
Let $K$ be a totally real abelian extension of $\mathbf Q$, $G$ its Galois group. For any positive integer $d$, put $\zeta_d = exp(2i\pi/d)$ and $\epsilon_{K,d} =N(1-\zeta_d)$, where $N$ is the norm from $\mathbf Q(\zeta_d)$ to $K \cap\mathbf Q(\zeta_d)$ . Sinnott's group $C_K$ is defined as the intersection of the group $U_K$ of units of $K$ with the sub-$\mathbf Z [G]$-submodule of $K^*$ generated by $-1$ and all the $\epsilon_{K,d}, d\in \mathbf N$. This coincides with Kummer's definition for $K= \mathbf Q(\zeta_p)^+$, and your ref.2 seems also to belong to Sinnott's game (I have not checked).  Of course Sinnott shows that $(U_K : C_K)$ is "almost" equal to the class number $h_K$, more precisely that $(U_K : C_K)=c_K .h_K$, where the constant $c_K$ is defined independently (otherwise, the formula would not make much sense!) and is "often" trivial. For example, if $p$ is a prime not dividing the degree of $K/\mathbf Q$ (the so called "semi-simple" situation), then the $p$-part of $c_K$ is trivial. In Iwasawa theory one frequently goes up the cyclomic tower $\cup K(\zeta_{p^n})$, which makes Sinnott's definition particularly well suited, because adapted to taking projective limits w.r.t. norms. One drawback is that Sinott's circular units do not behave smoothly under "Galois descent" (i.e. by taking fixed points under Galois action), contrary to Washington's, see below. They appear anyway to be the winners because of their crucial role in the algebraic part of the proof of the Main Conjecture of Iwasawa theory (now the theorem of Mazur-Wiles).
Washington's definition is more simply $W_K=K \cap C_{\mathbf Q(\zeta_d)}$, where $d$ is the conductor of $K$. It cetainly includes your ref.1. Obviously $W_K$ contains $C_K$, and your question is whether they coincide or not. As we said, for a given prime $p$ and in the semi-simple situation, the $p$-adified $(.)\otimes \mathbf Z_p$ coincide. But the general problem is not so obvious, and has been settled only around 2002 . In Iwasawa theory, one fixes a prime $p$ and take projective limits, up the cyclotomic tower, of the $p$-adified  circular units to get, say $\bar C_{\infty}$ and $\bar W_{\infty}$. A result of Belliard (2202) states that $\bar C_{\infty}=\bar W_{\infty}$ iff the $\Lambda$-module $\bar C_{\infty}$ is free, where $\Lambda$ denotes the Iwasawa algebra. Counter-examples to this $\Lambda$-freeness have been constructed by Belliard and Kucera, which settled the question. 
(*) At the beginning of your post, what do you mean by "cyclic abelian" ?
