Residue field

In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal.

This construction is applied in algebraic geometry, where to every point x of a scheme X one associates its residue field k(x). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point.