# Definition:Quotient Field/Definition 2

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## Definition

Let $D$ be an integral domain.

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