= Finite relative unit quotient over a real quadratic field
If $K$ is real quadratic and $L/K$ is a number-field extension with $\mathcal O_L^\times/\mathcal O_K^\times$ finite, then the unit ranks agree. If $d=[L:K]$ and $L$ has signature $(r_1,r_2)$, this forces
$$
r_1+r_2=2,\qquad r_1+2r_2=2d.
$$
For a proper extension the only possibility is $d=2$ and signature $(0,2)$. For $K=\mathbb Q(\sqrt5)$, the example $L=K(i)$ has this signature and a nontrivial finite quotient because $i$ is a new root of unity.
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