Definition:Field of Quotients
Definition
Let $D$ be an integral domain.
Let $F$ be a field.
Definition 1
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
- $(3): \quad \forall z \in F: \exists x \in D, y \in D_{\ne 0}: z = \dfrac {\map \iota x} {\map \iota y}$
Definition 2
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.
Definition 3
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
- $(3): \quad$ it satisfies the following universal property:
- For every field $E$ and for every ring monomorphism $\varphi: D \to E$, there exists a unique field homomorphism $\bar \varphi: F \to E$ such that $\varphi = \bar \varphi \circ \iota$
- That is, the following diagram commutes:
- $\begin{xy}\xymatrix@+1em@L+2px{D \ar[r]^\iota \ar[dr]_\varphi & F \ar[d]^{\exists_1 \bar \varphi} \\ & E}\end{xy}$
Definition 4
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 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 Field of Quotients.
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
- Existence of Field of Quotients, where it is shown that the field of quotients always exists
- Field of Quotients is Unique, which justifies the use of the definite article
- Results about fields of quotients can be found here.
Generalizations
Linguistic Note
The word quotient derives from the Latin word meaning how often.