Leopoldt's Conjecture for CM fields

TL;DR

This paper proves Leopoldt's conjecture for CM number fields using Iwasawa theory, Kummer theory, and class field theory, extending previous results known for abelian fields.

Contribution

It introduces a proof of Leopoldt's conjecture for CM fields, employing Iwasawa's methods and Takagi theory, which was previously unestablished.

Findings

Leopoldt's conjecture holds for all CM fields.

The proof combines Iwasawa theory with Kummer and class field theory.

This extends the class of fields for which the conjecture is verified.

Abstract

The conjecture of Leopoldt states that the $p$ - adic regulator of a number field does not vanish. It was proved for the abelian case in 1967 by Brumer, using Baker theory. We prove this conjecture for CM number fields $\K$. The proof uses Iwasawa's methods -- especially Takagi Theory -- for deriving his skew symmetric pairing, together with Kummer- and Class Field Theory.