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.^{[1]} 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*).^{[2]} 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.^{[clarification needed]}

Suppose that *R* is a commutative local ring, with 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*) of *x*, 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* at *x* is by definition the localization *A _{p}* of

*k*(*x*) :=*A*_{p}/*p*·*A*_{p}.

One can prove that this definition does not depend on the choice of the affine neighbourhood *U*.^{[3]}

A point is called *K*-rational for a certain field *K*, if *k*(*x*) = *K*.^{[4]}

Consider the affine line **A**^{1}(*k*) = Spec(*k*[*t*]) over a field *k*. If *k* is algebraically closed, there are exactly two types of prime ideals, namely

- (
*t*−*a*),*a*∈*k* - (0), the zero-ideal.

The residue fields are

- , the function field over
*k*in one variable.

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**.

- For a scheme locally of finite type over a field
*k*, a point*x*is closed if and only if*k*(*x*) is a finite extension of the base field*k*. This is a geometric formulation of Hilbert's Nullstellensatz. In the above example, the points of the first kind are closed, having residue field*k*, whereas the second point is the generic point, having transcendence degree 1 over*k*. - A morphism Spec(
*K*) →*X*,*K*some field, is equivalent to giving a point*x*∈*X*and an extension*K*/*k*(*x*). - 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.