Residue field

Remarks

The field $\kappa(\mathfrak p)$ is isomorphic to the field of fractions of the integral domain $R/\mathfrak p$.

Example: the residue field of $\Z_{(p)}$ is $\Z/p\Z$, since $a/b \mapsto a b^{-1} \bmod p$ is a surjection with kernel $p\Z_{(p)}$. The residue field of $k[\![x]\!]$ is $k$, by taking the constant term. For $R = \Z$, the residue field at $(0)$ is $\Q$ and the residue field at $p\Z$ is $\Z/p\Z$.

Non-example: in the local ring $\Z_{(p)}$, the quotient $\Z_{(p)}/p^2\Z_{(p)} \cong \Z/p^2\Z$ is not a field, because $p \ne 0$ there and $p \cdot p = 0$. Only the quotient by the maximal ideal is a field.