Residue field

In mathematics, the residue field is a basic construction in commutative algebra. If is a commutative ring and is a maximal ideal, then the residue field is the quotient ring, which is a field. Frequently, 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 of a scheme one associates its residue field. 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 is a commutative local ring, with maximal ideal. Then the residue field is the quotient ring.
Now suppose that is a scheme and is a point of. By the definition of a scheme, we may find an affine neighbourhood of, with some commutative ring. Considered in the neighbourhood, the point corresponds to a prime ideal . The local ring of at is by definition the localization of by, and has maximal ideal. Applying the construction above, we obtain the residue field of the point :
Since localization is exact, is the field of fractions of . One can prove that this definition does not depend on the choice of the affine neighbourhood.
A point is called -rational for a certain field, if.

Example

Consider the affine line over a field. If is algebraically closed, there are exactly two types of prime ideals, namely

, the zero-ideal.

The residue fields are

, the function field over k in one variable.

If is not algebraically closed, then more types arise from the irreducible polynomials of degree greater than 1. For example if, then the prime ideals generated by quadratic irreducible polynomials all have residue field isomorphic to.

Properties

For a scheme locally of finite type over a field, a point is closed if and only if is a finite extension of the base field. This is a geometric formulation of Hilbert's Nullstellensatz. In the above example, the points of the first kind are closed, having residue field, whereas the second point is the generic point, having transcendence degree 1 over.
A morphism, some field, is equivalent to giving a point and an extension.
The dimension of a scheme of finite type over a field is equal to the transcendence degree of the residue field of the generic point.