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It is known that there are finitely many square-free integers $d$ for which the ring of integers of $\mathbf{Q}(\sqrt{d})$ is Euclidean under the norm function.

Is there an analogous result for quadratic extensions of $K(t)$, $t$ an indeterminate and $K$ a (finite) field?

Also, can we classify the (imaginary) quadratic function fields that are PID in a fashion similar to the number field case?

I'm assuming the answers to these questions are well known, but my internet search has come up empty.

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  • $\begingroup$ This is just $L(t)$ for the finite field $L=K[\sqrt{D}]$ $\endgroup$
    – Aaron Meyerowitz
    Commented Jun 14, 2018 at 21:06
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    $\begingroup$ @AaronMeyerowitz $D$ may not be constant. $\endgroup$
    – Felipe Voloch
    Commented Jun 14, 2018 at 21:33
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    $\begingroup$ Queen, Clifford, Arithmetic euclidean rings. Acta Arith. 26 (1974/75), 105–113. $\endgroup$
    – Felipe Voloch
    Commented Jun 15, 2018 at 1:30
  • $\begingroup$ Obrigado, @FelipeVoloch. This is a nice reference, but it still does not provide a classification of the quadratic function fields that are euclidean with respect to the norm. Would you know if there's a reason to expect a finite number of those in this setting? $\endgroup$
    – rpc
    Commented Jun 18, 2018 at 15:20
  • $\begingroup$ Sorry, I don't have anything else to add. The obvious thing would be to see who cites Queen's paper to see if the case of Euclidean norms has been treated. It's possible that it's still open. Or maybe write to him. $\endgroup$
    – Felipe Voloch
    Commented Jun 18, 2018 at 22:11

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