---
tag: G51E8MW7J1
kind: concept
name: Residue field of a point
msc: [14A15]
specializes: [NJ3APQ8BHH]
sources:
  - url: https://en.wikipedia.org/wiki/Residue_field
---

## Summary

The residue field $\kappa(x)$ of a point $x$ of a [[scheme|5A35TYZD02]] is the [[residue field|NJ3APQ8BHH]] of the local ring of the scheme at $x$.
It is the field in which functions near $x$ take their "value at $x$".
Different points of one scheme can have residue fields of different characteristics, as for $\operatorname{Spec} \Z$.

## Explanations

### L1 human

On a variety over the complex numbers, the value of a polynomial at a point is a complex number.
On a scheme, the value of a function at a point lives in the residue field of that point, and this field can change from point to point.
On $\operatorname{Spec} \Z$, an integer $n$ has value $n \bmod p$ at the point $(p)$, in the field with $p$ elements.
At the point $(0)$ its value is $n$ itself, in the field $\Q$.