# Field Theory/Quotient fields

Definition (quotient field):

Let ${\displaystyle R}$ be an integral domain. The quotient field of ${\displaystyle R}$, often denoted ${\displaystyle \operatorname {Frac} (R)}$, is defined to be the field of formal fractions

${\displaystyle \left\{{\frac {a}{b}}|a\in R,b\in R^{\times }\right\}}$,

together with the multiplication and addition operations given by the formulae

${\displaystyle {\frac {a}{b}}{\frac {c}{c}}:={\frac {ac}{bd}}}$ and ${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}:={\frac {ad+cb}{bd}}}$.