Quotient Field is Unique

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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 (field) isomorphism $\phi: K \to L$ satisfying:

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


Proof

Follows directly from the Quotient Theorem for Monomorphisms.

$\blacksquare$


Comment

It follows from this result that 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 $D$.


Sources