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.
Read more about Residue Field: Definition, Example, Properties
Famous quotes containing the words residue and/or field:
“Every poem of value must have a residue [of language].... It cannot be exhausted because our lives are not long enough to do so. Indeed, in the greatest poetry, the residue may seem to increase as our experience increases—that is, as we become more sensitive to the particular ignitions in its language. We return to a poem not because of its symbolic [or sociological] value, but because of the waste, or subversion, or difficulty, or consolation of its provision.”
—William Logan, U.S. educator. “Condition of the Individual Talent,” The Sewanee Review, p. 93, Winter 1994.
“After all the field of battle possesses many advantages over the drawing-room. There at least is no room for pretension or excessive ceremony, no shaking of hands or rubbing of noses, which make one doubt your sincerity, but hearty as well as hard hand-play. It at least exhibits one of the faces of humanity, the former only a mask.”
—Henry David Thoreau (1817–1862)