Definition:Field of Quotients/Definition 4

Definition

Let $D$ be an integral domain.

A field of quotients of $D$ is a pair $\struct {F, \iota}$ which is its total ring of fractions, that is, the localization of $D$ at the nonzero elements $D_{\ne 0}$.

Also known as

Since the construction of the field of quotients $F$ from an integral domain $D$ 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

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