# 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}\[email protected][email protected]+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**.