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 (see Zariski topology). 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 (which is an integral domain as is a prime ideal)., 1.K One can prove that this definition does not depend on the choice of the affine neighbourhood .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.
A point is called -rational for a certain field , if .Görtz, Ulrich and Wedhorn, Torsten. Algebraic Geometry: Part 1: Schemes (2010) Vieweg+Teubner Verlag.
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 (such as ) 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.
See also
Arithmetic zeta function
References
Further reading , section II.2