Field of Quotients is Unique

Theorem

Let $\struct {D, +, \circ}$ be an integral domain.

Let $K, L$ be field of quotients of $\struct {D, +, \circ}$.

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

$\forall x \in D: \map \phi x = x$

Proof

Follows directly from the Quotient Theorem for Monomorphisms.

$\blacksquare$

Motivation

It follows from this result that when discussing an integral domain $\struct {D, +, \circ}$, all we need to do is select any particular field of quotients $K$ of $D$, and call $K$ the field of quotients of $D$.

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

This is also clearly the subfield of $L$ generated by $D$.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.10$: Corollary