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.[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.


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 pA (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 :

k(x) := Ap / p·Ap.

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 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 (x2 + 1) has residue field isomorphic to C.



  1. Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3 ed.). Wiley. ISBN 9780471433347.
  2. David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 3-540-63293-X.
  3. Intuitively, the residue field of a point is a local invariant. Axioms of schemes are set up in such a way as to assure the compatibility between various affine open neighborhoods of a point, which implies the statement.
  4. Görtz, Ulrich and Wedhorn, Torsten. Algebraic Geometry: Part 1: Schemes (2010) Vieweg+Teubner Verlag.

Further reading

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