View source for Field of fractions

{{TEX|done}}{{MSC|13}}

''of an [[integral domain]] $R$''; ''quotient field'', ''field of quotients''

The smallest field containing the integral domain $R$, considered as fractions with elements of $R$ as numerator and denominator.  The construction generalises the construction of the [[rational number]]s from the ring of integers.

Let $S$ denote $R \setminus {0}$: since $R$ is an integral domain, $S$ is closed under multiplication.  Define an equivalence $\sim$ on $R \times S$ by
$$
(x,y) \sim (u,v) \Leftrightarrow xv = yv \ .
$$
We denote the equivalence class of $(x,y)$ by $x/y$ and the quotient by $F$.  Operations are defined on $F$ by
$$
x/y + u/v = (xv+yu)/yv
$$
and
$$
x/y \cdot u/v = (xu)/(yv) \ .
$$
These definitions are compatible with the relation $\sim$, that is, do not depend on the choice of representative of the equivalence classes.

It may be verified that $F$ becomes a field under these operations, and the map from $R$ to $F$ by $x \mapsto x/1$ is an embedding.  The field of fractions has the [[universal property]] that if $R$ embeds in a field $K$ then the embedding extends to an embedding of $F$ into $K$.

In the opposite direction, given a field $F$, every subring $R$ of $F$ is necessarily an integral domain.  We can identify the field of fractions of $R$ with the set $\{ ab^{-1} : a \in R, b \in R\setminus\{0\}\}$ as a subfield of $F$.

See [[Fractions, ring of]] for more general constructions.

====References====
* P.M. Cohn, ''Skew Field Constructions'', London Mathematical Society lecture note series '''27''', Cambridge University Press (1977) {{ISBN|0-521-21497-1}}