Residue field

Let $(R,\mathfrak m)$ be a local ring. The residue field of $R$ is

k(R):=R/\mathfrak m.

It is a field because $\mathfrak m$ is maximal.

The canonical surjection

R \twoheadrightarrow R/\mathfrak m

is sometimes called the residue map.

More generally, for a prime ideal $\mathfrak p\subset R$ , the residue field at $\mathfrak p$ is

\kappa(\mathfrak p):=R_{\mathfrak p}/\mathfrak pR_{\mathfrak p},

where $R_{\mathfrak p}$ is the localization at $\mathfrak p$ . This is the residue field of the local ring $R_{\mathfrak p}$ .

Examples

$\mathbb Z_{(p)}$ . Its maximal ideal is $p\mathbb Z_{(p)}$ , and $\mathbb Z_{(p)}/p\mathbb Z_{(p)}\cong \mathbb F_p.$

$k[x]_{(x)}$ . Its maximal ideal is $(x)$ , so $k[x]_{(x)}/(x)\ \cong\ k$ via evaluation at $x=0$ .

$k[x,y]_{(x,y)}$ . Its maximal ideal is generated by $x$ and $y$ , and its residue field is $k$ .

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