I need to check true or false of the above statement. But unfortunately I haven't found any counter example yet. So if the statement is false, can anyone one give me a counterexample or if it is true, just give me a hint to prove it.

I know that $F[x]$ is a PID if and only if $F$ is a field. And every PID is an UFD. Please help me. Thanks.

  • 2
    $\begingroup$ In a field every nonzero nonunit vacuously has exactly one factorization. $\endgroup$ Aug 11, 2016 at 14:22
  • $\begingroup$ What does "Can we think in that direction if the statement is true" mean? $\endgroup$
    – user228113
    Aug 11, 2016 at 14:22

1 Answer 1


Every field $F$ is a UFD because it is an integral domain and it contains no primes -- everything non-zero is a unit -- so the requirement to be checked on factorization is vacuous.


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