This entry is about the notion of a residue field in algebraic geometry. There is another (related) notion of a residue field in constructive mathematics; see under field.
symmetric monoidal (∞,1)-category of spectra
Given a local ring $R$, by the definition there is a maximal ideal $\mathfrak{m}\subset R$. The quotient $R/\mathfrak{m}$ is therefore a division ring, and in commutative case, therefore a field, called the residue field. In algebraic geometry, the residue field at a point $x$ of a scheme $X$ is the residue field of the corresponding stalk $\mathcal{O}_{X,x}$ of the structure sheaf, which is by the definition a local ring.