Leopoldt’s conjecture in parameterized families

Abstract

For each fixed prime p ≠ 5 p \ne 5 , we prove Leopoldt’s conjecture in two infinite families of fields of degree five whose normal closure has Galois group over the rationals isomorphic to S 5 {S_5} . The units of these fields were determined by Maus [4]; we develop and apply a simple reformulation of Leopoldt’s conjecture to obtain the result. We also observe that Leopoldt’s conjecture in one field can imply the same in a second field related by congruence conditions.