Prove that if $R[x]$ is a PID, then $R$ is a field

Asked Apr 08 '16 at 13:44

Active May 07 '21 at 14:31

Viewed 3,522 times

I just need someone to check my proof and provide me feedback:

Since $R[x]$ is a PID, then the ideal $I = (x-1)$ generated by the polynomial $x-1$ is maximal because it is of degree 1 added to a constant.

So $R[x]/I\simeq R$ is a field, so $R$ is a field.

Is this proof enough/correct?

Thanks.

edited May 07 '21 at 14:31

glS

asked Apr 08 '16 at 13:44

Pacemaker

5

Sure, looks good (you could also just take the ideal generated by $x$). – Tobias Kildetoft Apr 08 '16 at 13:47
2

I don't think this is a duplicate. It is a different proof than the linked duplicate. – Peter_Pan Dec 15 '19 at 17:36

0 Answers0