Galois number fields: $r_1 = 0$ or $r_2 = 0$

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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 $K\to\mathbb C$). Statement. For every number field $K$ such that the extension $K/\mathbb Q$ is Galois (normal and separable), $r_1 = 0$ or $r_2 = 0$ (inclusive or).