"Let $R$ be an integral domain. Assuming that the Division Algorithm holds in $R[x]$, prove that $R$ is a field."

Asked Apr 17 '18 at 14:00

Active Apr 17 '18 at 14:04

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Let $R$ be an integral domain. Assuming that the Division Algorithm holds in $R[x]$, prove that $R$ is a field.

Can someone tell me if I am going about proving this correctly? My tactic was going to be to prove that every integral domain that works with the Division Algorithm (so I guess a Euclidean Domain) is finite, and thus would be a finite ring, which would make it a field based on the theorem that states just that. I don't know if this is valid logic, so could someone tell me if I'm at least on the right path?

edited Apr 17 '18 at 14:04

B. Mehta

13,032

asked Apr 17 '18 at 14:00

Eliza May

5

Welcome to MSE! Great first question. Observe that the integers are an integral domain and a euclidean domain, but they're not finite! – B. Mehta Apr 17 '18 at 14:03
What do you mean exactly with ‘works with division algorithm’? – Bernard Apr 17 '18 at 14:04
If $R[X]$ is a PID consider the ideal generated by $a\neq 0$ in $R$ and $X$... – marwalix Apr 17 '18 at 16:07

"Let $R$ be an integral domain. Assuming that the Division Algorithm holds in $R[x]$, prove that $R$ is a field."

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