Definition:Field of Quotients

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

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

Then $\left({F, \oplus, \cdot}\right)$ is the quotient field of $\left({D, +, \circ}\right)$ if:
 * $(1) \quad F$ contains an isomorphic copy of $D$
 * $(2) \quad $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 subring.

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.

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