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If R is an integral domain, show that the field of quotients Q is the smallest field containing R in the following sense: If R is a subset of F, where F is a field, show that F has subfield K such that R is a subset of K and K is isomorphic to Q.

I have trouble interpreting this question. My understanding is that we assume R is a subset of F, we want to prove that there exists K which is a subfield of F such that R is a subset of K and K is isomorphic to Q. That means we want to prove K is a subfield of F. Am I right? If I am right, then how to prove K is a subfield of F. Do I have to prove K is a subring first, then prove K is a field?

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    View $R$ as a subring of $Q$. Let $F$ be a field and $f:R\to F$ a monomorphism. Then there is a unique morphism $g:Q\to F$ which extends $f$.2012-03-25

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Hint: Suppose that $R\subseteq F$. Then, for all nonzero $x\in R$, $x\in U(F)=F\setminus\{0\}$. So, think about the field generated by the following subset of $K$: $\{xy^{-1}\,\vert\,x\in R, y\in R\setminus\{0\}\}$. Can you find an isomorphism from this field to $Q$?

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    Well, technically the [quotient field](http://en.wikipedia.org/wiki/Field_of_fractions) is a set of equivalence classes of $R\times (R\setminus\{0\})$.2012-03-25