Minkowski's theorem on independent conjugate units

TL;DR

This paper provides a new proof of Minkowski's theorem on the existence of special units in Galois extensions, establishing bounds on their heights and subgroup indices, with implications for relative units in intermediate fields.

Contribution

It offers a novel proof of Minkowski's theorem, demonstrating the existence of Minkowski units with height bounds and subgroup index estimates, including for relative units in intermediate Galois extensions.

Findings

Existence of Minkowski units with height comparable to fundamental units

Bounds on the index of conjugate units subgroup in the unit group

Analogous bounds for relative units in intermediate Galois extensions

Abstract

We call a unit $\beta$ in a Galois extension $l/\mathbb{Q}$ a Minkowski unit if the subgroup generated by $\beta$ and its conjugates over $\mathbb{Q}$ has maximum rank in the unit group of $l$. Minkowski showed the existence of such units in every Galois extension. We will give a new proof to Minkowski's theorem and show that there exists a Minkowski unit $\beta \in l$ such that the Weil height of $\beta$ is comparable with the sum of the heights of a fundamental system of units of $l$. Our proof implies a bound on the index of the subgroup generated by the algebraic conjugates of $\beta$ in the unit group of $l$. If $k$ is an intermediate field such that \begin{equation*} \mathbb{Q} \subseteq k \subseteq l, \end{equation*} and $l/\mathbb{Q}$ and $k/\mathbb{Q}$ are Galois extensions, we prove an analogous bound for the subgroup of relative units.