Leopoldt’s conjecture for the number field F

Prove that the Leopoldt defect δ(F) vanishes for the number field F, equivalently that the compositum of all Z_p-extensions of F has Galois group of rank r_2(F)+1 over F.

Background

The paper uses the Leopoldt defect δ(F) to describe the rank of the Galois group of the compositum of all Z_p-extensions of a number field F. Vanishing of this defect is the content of Leopoldt’s conjecture.

The conjecture is mentioned as a standing condition under which a particular S-ramified Z_p-extension is unique and as the assertion that the rank contribution δ(F) is zero. It is not proved in the paper.

References

It follows from Theorem 2.1 that the $\ZZ_p$-extension $F_{\infty}/F$ unramified outside $S$ exists, and it is unique if we assume that Leopoldt's conjecture holds for $F$.

— Pseudonullity for fine Selmer groups over multiple $\mathbb{Z}_p^{d}$-extensions  (2608.25309 - Qi et al., 26 Aug 2026) in Section 1, subsection “Preliminaries” and Section 1, subsection “Main results and tools”