Applications of representation theory and of explicit units to Leopoldt's conjecture

Let $L\mathrm{/}K$ be a Galois extension of number fields and let $G=\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}K)$. We show that under certain hypotheses on $G$, for a fixed prime number $p$, Leopoldt’s conjecture at $p$ for certain proper intermediate fields of $L\mathrm{/}K$ implies Leopoldt’s conjecture at $p$ for $L$. We also obtain relations between the Leopoldt defects of intermediate extensions of $L\mathrm{/}K$. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers $\mathcal{P}$, there exists an infinite family $\mathcal{F}$ of totally real $S_3$-extensions of $\mathbb{Q}$ such that Leopoldt’s conjecture for $F$ at $p$ holds for every $F\in \mathcal{F}$ and $p\in \mathcal{P}$.