Is ring of Gaussian rationals in unique factorization domain?

Question

Instead of Gaussian integers, let us think about Gaussian rationals, where $a$ and $b$ in $a+bi$ are rational numbers. Then would ring of Gaussian rationals be in unique factorization domain?

Answer 1

As you can also read in the comments to your question, the set of Gaussian rationals is a field. Seeing that a field is always a unique factorization domain, the answer to your question is yes!

Answer 2

I suspect that you meant to ask the less trivial question: does the unique factorization of elements of a UFD $D$ extend to elements of its quotient field $F,$ by allowing negative exponents on primes? Yes, since any nonunique factorization in $F$ would map to a nonunique factorization in $D,$ by scaling it to remove all negative powers.