Let $R$ be an [integral domain](/page/Integral%20Domain), with additive identity $0_R$ and multiplicative identity $1_R$. Then there exists a field $F$ together with an injective unital ring homomorphism $\iota: R \to F$ such that every element of $F$ has the form $\iota(a)\iota(b)^{-1}$ for some $a\in R$ and some $b\in R\setminus\{0_R\}$.

Moreover, if $(F,\iota)$ and $(F',\iota')$ are two such pairs, then there is a unique field isomorphism $\Phi: F \to F'$ such that $\Phi\circ \iota=\iota'$.