Residue field

Summary

The residue field of a local ring $R$ with maximal ideal $\mathfrak m$ is the quotient ring $R/\mathfrak m$, which is a field. More generally, the residue field of any commutative ring at a prime ideal $\mathfrak p$ is the residue field of the localization $R_{\mathfrak p}$. It is the field of "values at a point", and many invariants of local rings, such as the cotangent space, are vector spaces over it.