Applications of Baker Theory to the Conjecture of Leopoldt

TL;DR

This paper provides elementary proofs for specific cases of the Leopoldt conjecture using Baker theory and class field theory, focusing on solvable and totally real extensions where $p$ splits completely.

Contribution

It introduces simplified proofs for extreme cases of the Leopoldt conjecture, expanding the applicability of Baker theory and class field theory in number theory.

Findings

Elementary proof for solvable extensions

Elementary proof for totally real extensions with complete splitting

Sharpened methods from previous presentations

Abstract

In this paper we give a short, elementary proof of the following too extreme cases of the Leopoldt conjecture: the case when $\K/\Q$ is a solvable extension and the case when it is a totally real extension in which $p$ splits completely. The first proof uses Baker theory, the second class field theory. The methods used here are a sharpening of the ones presented at the SANT meeting in G\"ottingen, 2008 and exposed in \cite{Mi2}, \cite{Mi1}.