Definition:Field of Quotients/Definition 1

Definition

Let $D$ be an integral domain.

A field of quotients of $D$ is a pair $\struct {F, \iota}$ where:

$(1): \quad$ $F$ is a field

$(2): \quad$ $\iota : D \to F$ is a ring monomorphism

$(3): \quad \forall z \in F: \exists x \in D, y \in D_{\ne 0}: z = \dfrac {\map \iota x} {\map \iota y}$

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers