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We call a unit $$\beta $$ β in a finite Galois extension $$l/\mathbb {Q}$$ l/Q a Minkowski unit if the subgroup generated by $$\beta $$ β and its conjugates over $$\mathbb {Q}$$ 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 $$\beta \in l$$ β∈l such that the Weil height of $$\beta $$ 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 $$\beta $$ β in the unit group of l. If kis an intermediate field such that $$\mathbb {Q}\subseteq k \subseteq l$$ Q⊆k⊆l , and $$l/\mathbb {Q}$$ l/Q and $$k/\mathbb {Q}$$ 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.