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 $G=\text{Gal}(L/K)$ . 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 $P$ , 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 $F\in F$ and $p\in P$ .