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In exercise 1.2.11. of Bump's Automorphic Forms and Representations book, he deals with real quadratic fields $K$ in which $\mathcal{O}_K$ is generated (as a ring) by its norm-1 units. In this case one get a nice relationship between ideal classes and hyperbolic conjugacy classes. However, as Bump comments, this condition is not always satisfied (for real quadratic fields).

So my question is: for what number fields $K$ is $\mathcal{O}_K$ generated by the norm-1 units? Moreover, does being so generated have analogous implications for $K$ or its ideal class group?

Some obvious observations: it basically never happens for imaginary quadratic $K$, it always happens for cyclotomic $K$, and (I believe) it's pretty rare for real quadratic $K$.

Other observations and examples, obvious or not, are welcome in lieu of a complete answer.

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    $\begingroup$ For a real quadratic field of discriminant $d$, the condition holds if and only if $\frac{d}{4}+1$ is a square, so, as you said, rarely. However, the relationship Bump mentions works for all real quadratic fields, if you phrase it as, "there's a natural bijection between the ideal classes of all orders $\mathcal{O}$ of $K$ and the hyperbolic conjugacy classes of ${\rm PSL}_2(\mathbb{Z})$ with eigenvalues in $K$." Your condition guarantees that the ideals of the max order $\mathcal{O}_K$ correspond to the conjugacy classes having the fundamental unit as an eigenvalue. $\endgroup$
    – Gene S. Kopp
    Commented Oct 9, 2013 at 16:44

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