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

Abstract Let L / K be a Galois extension of number fields and let $$G=\textrm{Gal}(L/K)$$ G = Gal ( L / 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 / K implies Leopoldt’s conjecture at p for L . We also obtain relations between the Leopoldt defects of intermediate fields of L / 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 $$\mathcal {P}$$ P , there exists an infinite family $$\mathcal {F}$$ F of totally real $$S_{3}$$ S 3 -extensions of $$\mathbb {Q}$$ Q such that Leopoldt’s conjecture for F at p holds for every $$F \in \mathcal {F}$$ F ∈ F and $$p \in \mathcal {P}$$ p ∈ P .