Field of Fractions

The ring of fractions of an integral domain. The field of fractions of the ring of integers is the rational field , and the field of fractions of the polynomial ring over a field is the field of rational functions

$K(X_1,X_2,...,X_n)={(f(X_1,X_2,...,X_n))/(g(X_1,X_2,...,X_n)): f,g in K[X_1,X_2,...,X_n],g!=0}.$

The field of fractions of an integral domain is the smallest field containing , since it is obtained from by adding the least needed to make a field, namely the possibility of dividing by any nonzero element.