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A commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain.

I have done the part that if $F$ is a field then $F[x]$ is a PID using the division algorithm and contradicting the minimality of degree of a polynomial.

But I am facing difficulty to do the other part.

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    $\begingroup$ Suppose that $F$ is not a field, and choose some non-unit $a$. Consider the ideal $(a, x)$. $\endgroup$ – rogerl Mar 31 '15 at 18:04
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Suppose $k[X]$ is a PID. Prove that $(X)$ is a maximal ideal and then note $k\simeq k[X]/(X)$.

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