# Definition:Quotient Field

## Contents

## 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 $(F, \iota)$. 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 $\operatorname{Frac}(D)$, $Q \left({D}\right)$ and $\operatorname{Quot}(D)$.

## Also see

- Equivalence of Definitions of Quotient Field
- Existence of Quotient Field, where it is shown that the quotient field always exists. It is constructed by creating the inverse of every element of $D$ in a maximally efficient way.
- Quotient Field is Unique, which justifies the use of a definite article

### Generalizations

## Sources

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