  Let $A' = \varprojlim_n$ be the projective limit of the $p$-parts of the
ideal class groups of the $p$ integers in the $\Z_p$-cyclotomic extension
$\K_{\infty}/\K$ of a CM number field $\K$. We prove in this paper that the $T$
part $(A')^-(T) = 0$. This fact has been explicitly conjecture by Kuz'min in
1972 and was proved by Greenberg in 1973, for abelian extensions $\K/\Q$.
Federer and Gross had shown in 1981 that $(A')^-(T) = 0$ is equivalent to the
non-vanishing of the $p$-adic regulator of the $p$-units of $\K$.

Number Theory