Definition:Field of Quotients

Definition
Let $D$ be an integral domain.

Let $F$ be a field.

Also defined as
It is common to define a field of quotients simply as a field $F$, instead of a pair $\struct {F, \iota}$. The embedding $\iota$ is then implicit.

The field of quotients can also be defined to be the explicit construction from Existence of Quotient Field.

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 or fraction field of $D$.

Some sources prefer the term quotient field, but this can cause confusion with similarly named but unrelated concepts.

Common notations include $\map {\operatorname {Frac} } D$, $\map Q D$ and $\map {\operatorname {Quot} } D$.

Also see

 * Equivalence of Definitions of Field of Quotients
 * Existence of Quotient Field, where it is shown that the quotient field always exists
 * Quotient Field is Unique, which justifies the use of the definite article

Generalizations

 * Definition:Total Ring of Fractions
 * Definition:Localization of Ring