The conjecture of Leopoldt states that the $p$ - adic regulator of a number
field does not vanish. It was proved for the abelian case in 1967 by Brumer,
using Baker theory. A conjecture, due to Gross and Kuz'min will be shown here
to be in a deeper sense a dual of Leopoldt's conjecture with respect to the
Iwasawa involution. We prove both conjectures for arbitrary number fields $\K$.
The main ingredients of the proof are the Leopoldt reflection, the structure of
quasi - cyclic $\Z_p[ \Gal(\K/\Q) ]$ - modules of some of the most important
$\Lambda[ \Gal(\K/\Q) ]$ - modules occurring ($T$ acts on them like a constant
in $\Z_p$), and the Iwasawa skew symmetric pairing.