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 = 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 extensions 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, there exists an infinite family F of totally real S 3 -extensions of Q such that Leopoldt's conjecture for F at p holds for every F ∈ F and p ∈ P.Theorem 1.1. [Ax65, Bru67] Let K be a finite abelian extension of Q or of an imaginary quadratic field. Then Leo(K, p) holds for every prime number p.