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.
Definition.
Suppose that "R" is a commutative local ring, with the maximal ideal "m". Then the residue field is the quotient ring "R"/"m".
Now suppose that "X" is a scheme and "x" is a point of "X". By the definition of scheme, we may find an affine neighbourhood "U" = Spec("A"), with "A" some commutative ring. Considered in the neighbourhood "U", the point "x" corresponds to a prime ideal "p" ⊂ "A" (see Zariski topology). The "local ring" of "X" in "x" is by definition the localization "R" = "Ap", with the maximal ideal "m" = "p·Ap". Applying the construction above, we obtain the residue field of the point "x" :
One can prove that this definition does not depend on the choice of the affine neighbourhood "U".
A point is called "K"-rational for a certain field "K", if "k"("x") = "K".
Example.
Consider the affine line A1("k") = Spec("k"["t"]) over a field "k". If "k" is algebraically closed, there are exactly two types of prime ideals, namely
The residue fields are
If "k" is not algebraically closed, then more types arise, for example if "k" = R, then the prime ideal ("x"2 + 1) has residue field isomorphic to C.