Definition:Quotient Field

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Definition

Let $D$ be an integral domain.

Let $F$ be a field.

Definition 1

A quotient field of $D$ is a pair $(F,\iota)$ where:

$(1): \quad$ $F$ is a field
$(2): \quad$ $\iota : D \to F$ is a ring monomorphism
$(3): \quad \forall z \in F: \exists x \in D, y \in D_{\neq 0}: z = \dfrac {\iota \left({x}\right)} {\iota \left({y}\right)}$


Definition 2

A quotient field of $D$ is a pair $(F,\iota)$ where:

$(1): \quad$ $F$ is a field
$(2): \quad$ $\iota : D \to F$ is a ring monomorphism
$(3): \quad$ If $K$ is a field with $\iota \left({D}\right) \subset K \subset F$, then $K = F$.

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


Definition 3

A quotient field of $D$ is a pair $(F,\iota)$ 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 quotient field of $D$ is a pair $(F,\iota)$ 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 quotient field simply as a field $F$, instead of a pair $\struct {F, \iota}$. The embedding $\iota$ is then implicit.

The quotient field can also be defined to be the explicit construction from Existence of Quotient Field.


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, fraction field or field of quotients of $D$.

Common notations include $\map {\operatorname {Frac} } D$, $\map Q D$ and $\map {\operatorname {Quot} } D$.


Also see

  • Results about quotient fields can be found here.


Generalizations


Sources