• Open Neighborhoods in the Krull Topology
• The Krull Topology on Galois Groups
• Motivation: Why Topology?
• The Krull Topology
• Properties of the Krull Topology
• Continuity of the Galois Action
• Open and Closed Subgroups
• Key Takeaways

📐 Today’s Lesson: The Krull Topology on Galois Groups From: Galois Theory (Jacobson) — Section 24

function KrullTopologyVisualizer() { const [selectedSigma, setSelectedSigma] = useState(0);

const [neighborhoodIdx, setNeighborhoodIdx] = useState(0); const elem = elements[selectedSigma]; const nbhd = elem.neighborhoods[neighborhoodIdx];

§Open Neighborhoods in the Krull Topology

Select an element of Gal(ℚ̄/ℚ) and explore its open neighborhoods. Each neighborhood is a coset of the stabilizer of a finite extension.

Choose element

{elements.map((e, i) => (

setSelectedSigma(i); setNeighborhoodIdx(0); }} className=px-3 py-1.5 rounded text-sm font-medium transition-colors ${ selectedSigma === i ? ‘bg-cyan-600 text-white’ : ‘bg-dark-700 text-dark-300 hover:bg-dark-600’

{e.label}

))}

{elem.description}

Open neighborhoods (from larger to smaller)

{elem.neighborhoods.map((n, i) => ( setNeighborhoodIdx(i)} className=px-3 py-1.5 rounded text-sm font-medium transition-colors ${ neighborhoodIdx === i ? ‘bg-cyan-600 text-white’ : ‘bg-dark-700 text-dark-300 hover:bg-dark-600’

{n.field}

))}

Finite extension: {nbhd.field}

Coset: {nbhd.coset}

{nbhd.description}

As the finite extension grows larger, the neighborhood shrinks (the coset contains fewer elements). The collection of all such cosets forms a basis for the Krull topology.

);

§The Krull Topology on Galois Groups

For finite Galois extensions, the Galois group is a finite group and the fundamental theorem works without any topological considerations. For infinite Galois extensions, the situation is more subtle: the Galois group is infinite, and the fundamental theorem holds only when we equip it with the correct topology – the Krull topology. This section develops this topology and its basic properties.

§Motivation: Why Topology?

Consider the Galois extension ℚ̄/ℚ. Its Galois group G = Gal(ℚ̄/ℚ) is an uncountable group. The fundamental theorem for finite extensions says: subgroups of Gal(L/K) correspond bijectively to intermediate fields. For infinite extensions, this fails if we take all subgroups. The correction is to restrict to closed subgroups in the Krull topology.

✏️ Example — The problem with arbitrary subgroups

The group G = Gal(ℚ̄/ℚ) has uncountably many subgroups, but there are only countably many subfields of ℚ̄ containing ℚ (since ℚ̄ is countable). So there cannot be a bijection between all subgroups and intermediate fields. We need to identify which subgroups are “visible” to the field structure.

§The Krull Topology

📖 Definition — The Krull topology

Let E/K be a (possibly infinite) Galois extension with group G = Gal(E/K). The Krull topology on G is defined by declaring the following sets to be a basis of open neighborhoods of any σ ∈ G:

{σ · Gal(E/F) : F/K is a finite Galois subextension of E/K}.

Equivalently, a net σᵢ → σ in the Krull topology if and only if for every finite Galois subextension F/K, eventually σᵢ|_F = σ|_F.

In words: two automorphisms are “close” in the Krull topology if they agree on a large finite subextension. The basic open sets are cosets of the stabilizers Gal(E/F) for finite Galois subextensions F/K.

§Properties of the Krull Topology

📐 Theorem — Theorem (Topological properties of the Galois group)

Let G = Gal(E/K) with the Krull topology. Then:

• G is a topological group: multiplication G × G → G and inversion G → G are continuous. • G is Hausdorff: if σ ≠ τ, there exists a finite F/K with σ|_F ≠ τ|F. • G is compact: this follows from Tychonoff’s theorem since G embeds as a closed subset of Π_F/_K Gal(F/K) (a product of finite discrete groups). • G is totally disconnected: the connected component of any element is just that element.

Property (3) deserves emphasis. We can write

G = Gal(E/K) ↪ Π_F Gal(F/K),

where the product ranges over all finite Galois subextensions F/K of E/K. Each Gal(F/K) is finite and discrete, so the product is compact by Tychonoff. The image of G is closed (it is defined by the compatibility conditions: restrictions must be consistent), so G is compact.

Compact + Hausdorff + totally disconnected = the definition of a profinite topological space. Thus Galois groups of infinite extensions are always profinite. The converse is also true: every profinite group arises as a Galois group (a deep result of Waterhouse and others).

§Continuity of the Galois Action

📐 Theorem — Proposition (Continuity of the Galois action)

The natural action of G = Gal(E/K) on E (where E is given the discrete topology) is continuous. That is, for each α ∈ E, the map σ ↦ σ(α) from G → E is continuous.

Proof. For any α ∈ E, let F be the normal closure of K(α)/K inside E. Then F/K is a finite Galois extension, and σ(α) depends only on σ|_F. The preimage of {σ(α)} is the coset σ · Gal(E/F), which is open.

§Open and Closed Subgroups

📐 Theorem — Proposition (Open subgroups)

A subgroup H ≤ G is open in the Krull topology if and only if H contains Gal(E/F) for some finite Galois extension F/K. Equivalently, H is open if and only if [G : H] < ∞.

Since G is compact, every open subgroup has finite index. Conversely, in a compact group, every subgroup of finite index is open (and closed). Thus the open subgroups are precisely the finite-index subgroups.

📐 Theorem — Proposition (Closed subgroups and fixed fields)

A subgroup H ≤ G is closed if and only if H = Gal(E/E^H), where E^H = {x ∈ E : σ(x) = x for all σ ∈ H} is the fixed field. In other words, H is closed if and only if it is “recovered” from its fixed field.

✏️ Example — Example: The closure of a subgroup

If H ≤ G is not closed, then its closure H̄ in the Krull topology is Gal(E/E^H). We always have E^H = E^H^̄, so the fixed field of H and H̄ are the same. Different non-closed subgroups can have the same fixed field, which is precisely why the Galois correspondence fails for non-closed subgroups.

§Key Takeaways

• 1.

Infinite Galois groups require the Krull topology for the fundamental theorem to work. Without topology, the bijection between subgroups and intermediate fields fails. • 2.

The Krull topology is defined by the basis of cosets of stabilizers of finite Galois subextensions. Two automorphisms are “close” if they agree on a large finite piece. • 3.

Gal(E/K) with the Krull topology is compact, Hausdorff, and totally disconnected (i.e., profinite). • 4.

Open subgroups correspond to finite extensions; closed subgroups are those recovered from their fixed fields. • 5.

The Galois action on E is continuous when E has the discrete topology, because each element belongs to a finite Galois subextension.

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