Is this obvious? I cannot see that this is true. The converse is fairly obvious though. I tried to show $(x)$ is a maximal ideal and try the quotient but failed. I will appreciate any help.
2026-03-28 01:15:28.1774660528
If $R$ is an integral domain and $R[x]$ is an euclidean domain, then $R$ is a field
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More generally, $R[x]$ is a PID iff $R$ if a field.
Hint: Take $r \in R \setminus 0$ and consider the ideal $(r,x)$.