Minkowski's theorem on independent conjugate units

Abstract

We call a unit $β$ in a Galois extension $l/\mathbb{Q}$ a Minkowski unit if the subgroup generated by $β$ 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 $β\in l$ such that the Weil height of $β$ 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 $β$ 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.