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It seems obvious that the Gaussian rational numbers, $G = \{a+bi : a, b\in Q\}$, are a field after running through the properties of a field (commutativity, associativity, identities, additive inverse, multiplicative inverse, distributive property). The problem I was working on asked if the Gaussian rationals form a field and let me assume that the complex numbers are a field.

It didn't say to assume the rationals are a field, but I wasn't sure if that's just because it's obvious, or if the instructor explicitly wanted you to only assume the complex numbers are a field.

For the multiplicative inverse property, I was thinking I could basically say that any $x\in G$ has an inverse because such an $x$ is an element of $C$ as well. My question is, can I just say $x^{-1}\in C$ is its inverse?

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    You need to *prove* that the inverse lies in $G$. Of course that is easy by rationalizing the denominator (or, equivalently, taking norms, or rearranging the minimal polynomial, etc). – 2017-02-03
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    $\frac{1}{a+ib} = \frac{a-ib}{a^2+b^2}$ so the commutative ring $\mathbb{Q}(i)$ is a field – 2017-02-03

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