Definition:Field of Quotients/Definition 2

Definition

Let $D$ be an integral domain.

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

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

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

$(3): \quad$ If $K$ is a field with $\iota \sqbrk D \subset K \subset F$, then $K = F$.

That is, the field of quotients of an integral domain $D$ is the smallest field containing $D$ as a subring.

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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

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras