In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and is a maximal ideal, then the residue field is the quotient ring k =, which is a field. Frequently, R is a local ring and is then its unique maximal ideal. In abstract algebra, the splitting field of a polynomial is constructed using residue fields. Residue fields also 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 maximal ideal. Then the residue field is the quotient ring. 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 of x, with some commutative ring A. Considered in the neighbourhood \mathcal{U}, the point x corresponds to a prime ideal (see Zariski topology). The local ring of X at x is by definition the localization of A by, and has maximal ideal =. 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 \mathcal{U}. A point is called -rational for a certain field k, if k(x)=k.

Example

Consider the affine line 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, then the prime ideal (x^2+1) has residue field isomorphic to \mathbb{C}.

Properties