Residue field

In mathematics, the residue field is a basic construction in commutative algebra.

is a commutative ring and

is a maximal ideal, then the residue field is the quotient ring

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

one associates its residue field

[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

is a commutative local ring, with maximal ideal

Then the residue field is the quotient ring

By the definition of a scheme, we may find an affine neighbourhood

, with some commutative ring

Considered in the neighbourhood

corresponds to a prime ideal

(see Zariski topology).

The local ring of

is by definition the localization

has maximal ideal

Applying the construction above, we obtain the residue field of the point

: One can prove that this definition does not depend on the choice of the affine neighbourhood

[3] A point is called

[4] Consider the affine line

is algebraically closed, there are exactly two types of prime ideals, namely The residue fields are If

is not algebraically closed, then more types arise, for example if

, then the prime ideal

has residue field isomorphic to