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.

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.