Residue field

In mathematics, the residue field is a basic construction in commutative algebra. If ${\displaystyle R}$ is a commutative ring and ${\displaystyle {\mathfrak {m}}}$ is a maximal ideal, then the residue field is the quotient ring ${\displaystyle k=R/{\mathfrak {m}}}$, which is a field. Frequently, ${\displaystyle R}$ is a local ring and ${\displaystyle {\mathfrak {m}}}$ 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 ${\displaystyle x}$ of a scheme ${\displaystyle X}$ one associates its residue field ${\displaystyle k(x)}$. 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.