Field of Fractions
The ring of fractions of an integral domain. The field of fractions of the ring of integers Z is the rational field Q, and the field of fractions of the polynomial ring K[X_1,X_2,...,X_n] over a field K is the field of rational functions
| [画像: 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}. ] |
![画像: 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}.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FieldofFractions/NumberedEquation1.svg){kind=link}
The field of fractions of an integral domain R is the smallest field containing R, since it is obtained from R by adding the least needed to make R a field, namely the possibility of dividing by any nonzero element.
See also
Localization, Ring of Fractions, Total Ring of FractionsThis entry contributed by Margherita Barile
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Cite this as:
Barile, Margherita. "Field of Fractions." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FieldofFractions.html