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 $\varphi = \bar \varphi \circ \iota$
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 $F$ 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 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
Sources
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- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras