Field of Quotients is Unique

Theorem
Let $$\left({D, +, \circ}\right)$$ be an integral domain.

Let $$K, L$$ be quotient fields of $$\left({D, +, \circ}\right)$$.

Then there is one and only one isomorphism $$\phi: K \to L$$ satisfying:


 * $$\forall x \in D: \phi \left({x}\right) = x$$

Proof
Follows directly from the Quotient Theorem for Monomorphisms.

Comment
So, when discussing an integral domain $$\left({D, +, \circ}\right)$$, all we need to do is select any particular quotient field $$K$$ of $$D$$, and call $$K$$ the quotient field of $$D$$.

If $$D$$ is already a subdomain of a specified field $$L$$, then the quotient field selected will usually be the subfield of $$L$$ consisting of all the elements $$x / y$$ where $$x \in D, y \in D^*$$ (see Quotient Field of Subdomain.) This is also clearly the subfield of $$L$$ generated by $$D$$.