Minkowski’s theorem on independent conjugate units

We call a unit β in a finite Galois extension l/ Q a Minkowski unit if the subgroup generated by β and its conjugates over Q has maximum rank in the unit group of l. Minkowski showed the existence of such units in every Galois extension. We give a new proof of Minkowski’s theorem and show that there exists a Minkowski unit β∈ l such that the Weil height of β is comparable with the sum of the heights of a fundamental system of units for 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 Q⊆ k⊆ l, and l/ Q and k/ Q are Galois extensions, we prove an analogous bound for the subgroup of relative units. In order to establish our results for relative units, a number of new ideas are combined with techniques from the geometry of numbers and the Galois action on places.