Definition:Field of Quotients

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

Let $\left({F, \oplus, \cdot}\right)$ be a field.

Then $\left({F, +, \circ}\right)$ is the quotient field of $\left({D, +, \circ}\right)$ if;
 * 1) $F$ contains an isomorphic copy of $D$
 * 2) If $\left({\tilde F, \tilde \oplus, \tilde \cdot}\right)$ is another field satisfying 1. and $\tilde F \subseteq F$, then $F = \tilde F$.

That is, the quotient field of an integral domain $\left({D, +, \circ}\right)$ is the minimal field containing $D$ as a subset.

Uniqueness is immediate from part 2. of the definition.

By Existence of Quotient Field, the quotient field always exists, and is constructed by inverting every element of $\left({D, +, \circ}\right)$ in a maximally efficient way.

By Quotient Field is Unique this construction satisfies the uniqueness criterion given by part 2. of the definition.

Universal Property
The quotient field can alternatively be defined by the following universal property:

Let $F$ be a field, and $\iota : D \to F$ an injective homomorphism such that for every field $\tilde F$ and for every injective homomorphism $\phi : D \to \tilde F$, there exists a unique homomorphism $\psi : F \to \tilde F$ satisfying $\psi \iota = \phi$. That is, the following diagram commutes:


 * FieldFracComDiag.jpg

Then $F$ is unique up to unique isomorphism, and is the quotient field of $D$.

Note
Since the construction of $F$ from $D$ mirrors the construction of the rationals from $\Z$, $F$ is sometimes called the field of fractions of $D$.

Common notations include $F = Q(D)$ and $F = \operatorname{Frac}D$.