Krull topology

Definition

Let $L/K$ be a Galois extension, possibly of infinite degree, with Galois group $G = \Gal(L/K)$ . Let $\mathcal{E}$ be the set of intermediate fields $E$ of $L/K$ with $[E : K]$ finite. The Krull topology on $G$ is the following topology. A subset $U \subseteq G$ is open if and only if for each $\sigma \in U$ there is $E \in \mathcal{E}$ with $\sigma \Gal(L/E) \subseteq U$ . Equivalently, the cosets $\sigma \Gal(L/E)$ , for $\sigma \in G$ and $E \in \mathcal{E}$ , form a basis of the topology.

Remarks

With this topology $G$ is a topological group that is compact, Hausdorff and totally disconnected, that is, a profinite group. In fact $G$ is isomorphic, as a topological group, to the inverse limit of the finite groups $\Gal(E/K)$ over the finite Galois subextensions $E/K$ . Example: if $L/K$ is finite, then $L \in \mathcal{E}$ and $\Gal(L/L) = \set{\id}$ , so every subset of $G$ is open and the topology is discrete. Non-example of an open set: for $L$ an algebraic closure of $\F_p$ , the one-point set $\set{\id}$ is not open. Indeed it would contain some $\Gal(L/\F_{p^n})$ , which is infinite since $L/\F_{p^n}$ has infinite degree.

Tag	`V7ERB5XHQR`
URI	`m2://V7ERB5XHQR`
Kind	definition
Writing	W0 stub
MSC	12F10, 20E18
Defines	Krull topology
Uses	Galois extension, Galois group, Intermediate field, Topological space, Compact space, Hausdorff space, Profinite group
Changed	2 November 2026

Tag

V7ERB5XHQR

URI

m2://V7ERB5XHQR

Kind

definition

Writing

W0 stub

MSC

12F10, 20E18

Defines

Krull topology

Uses

Galois extension, Galois group, Intermediate field, Topological space, Compact space, Hausdorff space, Profinite group

Changed

2 November 2026

Krull topology

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here