  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. We prove this conjecture for CM number fields $\K$. The
proof uses Iwasawa's methods -- especially Takagi Theory -- for deriving his
skew symmetric pairing, together with Kummer- and Class Field Theory.
