# How to prove a number system is a fraction field of another?

For example, how can I show that $\mathbb{Q}$ is the fraction field of $\mathbb{Z}$? Or that $\mathbb{C}$ is the fraction field of $\mathbb{R}$?

I understand that $\mathbb{Z}$ is a subring of $\mathbb{Q}$ & each r in $\mathbb{Q}$ can be written as a fraction r = a/b with a,b in $\mathbb{Z}$ and no proper subfield of $\mathbb{Q}$ has that property. But is there some general way to show this for the other number systems?

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The universal mapping property of localizations (or fraction fields) yields an easy test for isomorphism, see the Corollary below, from Atiyah & MacDonald, Commutative Algebra, p. 39.

In your case $\,A\,$ is a domain and $\,S\,$ is the set of nonzero elements in $A.\,$ Since $\,S\,$ contains no zero-divisors, condition $(ii)$ in the Corollary simplifies to $\,g:A\to Q\,$ is an injection. So $\,B = Q\,$ is isomorphic to the quotient field of $A\,$ if $\,Q\,$ contains an isomorphic image $\bar A$ of $A$ such every nonzero $\,a\in A\,$ maps to a unit $\, \bar a = g(a)\,$ in $Q,\,$ and every $\,q\in Q\,$ is a fraction over $\bar A,\,$ i.e. $\,q = \bar a/\bar b = \bar a \bar b^{-1}\,$ for some $\,0\neq b,\,a\in A.$