True or false: Every Field is a UFD.

Question

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.

In a field every nonzero nonunit vacuously has exactly one factorization. — hmakholm left over Monica, Aug 11 '16 at 14:22
What does "Can we think in that direction if the statement is true" mean? — , Aug 11 '16 at 14:22

score 19 · Accepted Answer · answered Aug 11 '16 at 14:23

19

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.

answered Aug 11 '16 at 14:23

Mr. Chip

5,009

True or false: Every Field is a UFD.

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