Some remarks on the $l$-adic Dirichlet theorem and the $l$-adic regulator

The Leopoldt conjecture is proved for some special types of nonabelian number fields. A new definition of the $l$-adic regulator is proposed, which makes sense for any algebraic number field, and in some special cases it is proved that this regulator is different from zero. In addition, estimates are obtained for the rank of the matrix figuring in the definition of the regulator.
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