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Question: What are the (noetherian or in characteristic 0 if needed) principal ideal domains that have finitely many units?

Can such rings be classified?

(This is a more specialised version of the question in Integral domains with finitely many units and got split off of this thread)

The well known examples are the imaginary quadratic integer rings of $\mathbb{Q} (\sqrt{d})$ for $d \in \{−1, −2, −3, −7, −11, −19, −43, −67, −163 \}$.

Is there a nice infinite family in characteristic 0 that are noetherian?

Question: What are the (in characteristic 0 if needed) principal ideal domains that have finitely many units?

Can such rings be classified?

(This is a more specialised version of the question in Integral domains with finitely many units and got split off of this thread)

The well known examples are the imaginary quadratic integer rings of $\mathbb{Q} (\sqrt{d})$ for $d \in \{−1, −2, −3, −7, −11, −19, −43, −67, −163 \}$.

Is there a nice infinite family in characteristic 0?

Question: What are the (noetherian+separable or in characteristic 0 if needed) principal ideal domains that have finitely many units?

Can such rings be classified?

(This is a more specialised version of the question in Integral domains with finitely many units and got split off of this thread)

The well known examples are the imaginary quadratic integer rings of $\mathbb{Q} (\sqrt{d})$ for $d \in \{−1, −2, −3, −7, −11, −19, −43, −67, −163 \}$.

Is there a nice infinite family in characteristic 0 that are noetherian?

Question: What are the (noetherian or in characteristic 0 if needed) principal ideal domains that have finitely many units?

Can such rings be classified?

(This is a more specialised version of the question in Integral domains with finitely many units and got split off of this thread)

The well known examples are the imaginary quadratic integer rings of $\mathbb{Q} (\sqrt{d})$ for $d \in \{−1, −2, −3, −7, −11, −19, −43, −67, −163 \}$.

Is there a nice infinite family in characteristic 0 that are noetherian?

Question: What are the (noetherian+seperable or in characteristic 0 if needed) principal ideal domains that have finitely many units?

Can such rings be classified?

(This is a more specialised version of the question in Integral domains with finitely many units and got split off of this thread)

The well known examples are the imaginary quadratic integer rings of $\mathbb{Q} (\sqrt{d})$ for $d \in \{−1, −2, −3, −7, −11, −19, −43, −67, −163 \}$.

Is there a nice infinite family in characteristic 0 that are noetherian?

Question: What are the (noetherian+separable or in characteristic 0 if needed) principal ideal domains that have finitely many units?

Can such rings be classified?

(This is a more specialised version of the question in Integral domains with finitely many units and got split off of this thread)

The well known examples are the imaginary quadratic integer rings of $\mathbb{Q} (\sqrt{d})$ for $d \in \{−1, −2, −3, −7, −11, −19, −43, −67, −163 \}$.

Is there a nice infinite family in characteristic 0 that are noetherian?