TITLE: A question on definition of field of fractions
QUESTION [11 upvotes]: Wikipedia defines the field of fractions of a domain as

The field of fractions or field of quotients of an integral domain is the "smallest" field in which it can be embedded.

What does "smallest" mean mathematically in this context? Is it possible to embed an integral domain in two different fields which have no elements in common?

REPLY [8 votes]: For completeness why don't we go ahead and include a proof of the universal property.
Let $R$ be an integral domain, $K$ its field of fractions and $\varphi: R \rightarrow F$ in an injective homomorphism into some field $F$. Consider the map $\tilde{\varphi}: K \rightarrow F$ given by $\tilde{\varphi}(a/b)=\varphi(a)/\varphi(b)$. We claim this map is in fact also an injective homomorphism. First note that the map is well defined because $F$ is a field so each element in $F$ has an inverse. Since $\varphi$ is a homomorphism $\tilde{\varphi}$ takes $0_K$ to $0_F$ and $1_K$ to $1_F$. To see additivity we have $\newcommand{\tv}{\tilde{\varphi}}$
$$\tv\left(\frac{a}{b}+\frac{c}{d}\right)=\tv\left(\frac{ad+bc}{bd}\right)
=\frac{\varphi(ad)+\varphi(bc)}{\varphi(bd)}=\frac{\varphi(a)}{\varphi(b)}+\frac{\varphi(c)}{\varphi(d)}=\tv\left(\frac{a}{b}\right)+\tv\left(\frac{c}{d}\right).$$
Multiplication is even more straightforward 
$$\tv\left(\frac{ac}{bd}\right)=\frac{\varphi(a)\varphi(c)}{\varphi(b)\varphi(d)}=\tv\left(\frac{a}{b}\right)\tv\left(\frac{c}{d}\right).$$
Injectivity follows immediately because $K$ is a field. Finally $\tv$ is the unique extension of $\varphi$. Let $\psi: K \rightarrow F$ be another extension of $\varphi$, that is $\psi|_R=\varphi$. Then for $b \in R \setminus \{0\}$
$$1=\psi(b/b)=\psi(b)\psi(1/b)=\varphi(b)\psi(1/b)$$
so $\psi(1/b)=1/\varphi(b)$ and we see that $\psi(a/b)=\tv(a/b)$.