The unit rank is positive except for $\mathbb Q$ and imaginary quadratic fields

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Throughout, $K$ is a number field (a field of characteristic zero that is finite-dimensional over $\mathbb Q$), $\mathcal O_K$ is its ring of integers and $\mathcal O_K^\times$ the unit group. $r_1$ denotes the number of real infinite places of $K$ and $r_2$ the number of complex infinite places (each complex place corresponds to a pair of complex-conjugate embeddings $K\to\mathbb C$). The rank of an abelian group $A$ is its rank as a $\mathbb Z$-module: the supremum of the cardinalities of $\mathbb Z$-linearly independent families (for the multiplicative group $\mathcal O_K^\times$: multiplicatively independent families). Torsion elements never belong to such a family. Statement. For every number field $K$: the rank of $\mathcal O_K^\times$ (a cardinal) is strictly greater than $0$ if and only if it is not the case that either $[K:\mathbb Q]=1$, or ($[K:\mathbb Q]=2$ and $K$ is totally complex, i.e. every infinite place of $K$ is complex).