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

Let L / K be a Galois extension of number fields and let G=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 P P , there exists an infinite family F F of totally real S₃ S 3 -extensions of Q Q such that Leopoldt’s conjecture for F at p holds for every F ∈ F F ∈ F and p ∈ P p ∈ P .