Definition
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 p ⊂ A (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.
A point is called K-rational for a certain field K, if k(x) ⊂ K.
Read more about this topic: Residue Field
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