Geodesic
Some remarks on Leopoldt's conjecture
Algebra i analiz, Tome 10 (1998) no. 6, pp. 144-155.

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Let $k$ be a number field, and let $p$ be a fixed prime number. Then the vanishing of the Leopoldt kernel $\mathscr{L}_p(k)$ is shown to be equivalent to the validity of a "Strong Local-Global Principle on units of $k$". This adds a problem of effectivity to Leopoldt's conjecture (an example to which is provided by the classical Kummer lemma on the $p$th powers of units in the field of the $p$th roots of unity). Some further remarks pertain to $\mathscr{L}_p(k)$ as a Galois module. For example, if $k/{\mathbb Q}$ is an Abelian $p$-extension, then the triviality of $\mathscr{L}_p(k)$ can be shown quite easily (in particular, without using Brumer's transcendency theorem). 
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     author = {F. Lorenz},
     title = {Some remarks on {Leopoldt's} conjecture},
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     number = {6},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_1998_10_6_a3/}
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F. Lorenz. Some remarks on Leopoldt's conjecture. Algebra i analiz, Tome 10 (1998) no. 6, pp. 144-155. http://geodesic.mathdoc.fr/item/AA_1998_10_6_a3/