Definition:Field of Quotients

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

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

Definition 1
$F$ is the quotient field of $D$ :
 * $(1): \quad$ There exists an embedding $\iota: D \to F$
 * $(2): \quad$ $F$ satisfies the following universal property:
 * For every field $E$ and for every embedding $\varphi : D \to E$, there exists a unique field homomorphism $\bar \varphi : F \to E$ such that $\varphi = \bar\varphi \circ \iota$

Definition 2
$F$ is the quotient field of $D$ :
 * $(1): \quad$ There exists an embedding $\iota : D \to F$
 * $(2): \quad$ $\displaystyle \forall z \in F: \exists x \in D, y \in D^\times: z = \frac{\iota (x)}{\iota (y)}$

Definition 3
$\left({F, \oplus, \cdot}\right)$ is the quotient field of $\left({D, +, \circ}\right)$ if:
 * $(1): \quad$ There exists an embedding $\iota : D \to F$
 * $(2): \quad$ If $K$ is a field with $\iota(D) \subset K \subset F$, then $K=F$.

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

Also known as
Since the construction of $F$ from $D$ mirrors the construction of the rationals from $\Z$, $F$ is sometimes called the field of fractions, fraction field or field of quotients of $D$.

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

Also defined as
The quotient field may also be defined as the construction in Existence of Quotient Field, though such concretization has little benefit.

Also see
By Existence of Quotient Field, the quotient field always exists, and is constructed by creating the inverse of 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.