Residue field ($\kappa (\mathfrak{m})=R/\mathfrak{m}$)

The residue field at a maximal ideal $\mathfrak{m}$ is the quotient field $\kappa (\mathfrak{m})=R/\mathfrak{m}$. At a prime ideal $\mathfrak{p}$, the residue field is the fraction field of $R/\mathfrak{p}$.

A local homomorphism of local rings $(R,\mathfrak{m})\to (S,\mathfrak{n})$ induces the residue-field extension $R/\mathfrak{m}\to S/\mathfrak{n}$. For an extension of non-Archimedean local fields, its degree is the residue-field degree.

The residue characteristic of a local ring is the characteristic of a ring of its residue field. A finite extension of ${\mathbb{Q}}_p$ has residue characteristic $p$.