It is well-known that $R[x]$ is a PID iff $R$ is a field. Is there a necessary and sufficient condition on $R$ for $R[x]$ to be a Euclidean domain?
2026-03-27 05:56:13.1774590973
When is $R[x]$ a Euclidean domain?
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I think you've answered your own question! If $R$ is a field, then $R[x]$ is a Euclidean domain. Conversely, if $R[x]$ is a Euclidian domain, then $R[x]$ is a PID, so $R$ is a field.