Math 250a: Higher Algebra
Problem Set #3 (13 October 2004): Infinite Galois theory
(and a bit of Kummer theory)
A field extension L/F , not necessarily finite, is said to be normal or Galois if every a∈Lis a
root of a separable polynomial in F[X] that splits completely in L. Examples are any finite Galois
extension, the extension of Qgenerated by all roots of unity, an algebraic closure of a perfect field,
or a separable closure of an arbitrary field. The Galois group G= Gal(L/F ) of such an extension
is defined as in the finite case: the group of all automorphisms of L/F , i.e., all automorphisms η
of Lsuch that η(c) = cfor all c∈F. This group carries a topology T, that is, a distinguished
collection of subsets called “open sets”. A subset S⊆Gis said to be “open” if for each η∈Sthe
field Lcontains a field E, of finite dimension over F, such that η0∈Sfor all η0that agree with η
on E, i.e., such that η0(x) = η(x) for all x∈E. (Equivalently:
NE(η) := {η0∈G| ∀x∈E, η0(x) = η(x)}
is a neighborhood of η, and the family of such neighborhoods as E , η vary is a family of basic open
sets for T.) Note that we may assume that E/F is normal, because the normal closure of any finite
E/F is again finite, and contained in Lif Eis.
1. Verify that Tis indeed a topology, that is, that Tcontains ∅and Gand is closed under finite
intersections and arbitrary unions. Prove that Gis a topological group for T, that is, that
the group operations (inverse and product) on Gare continuous. Check that if [L:F]<∞
then Tis the discrete topology (all subsets are open).
2. Let Kbe any subfield of Lcontaining F. Prove that Gal(L/K) is a closed subgroup of Gal(L/F ),
and is a normal subgroup if K/F is normal. If His any normal subgroup of Gal(L/F), prove
that g(LH) = LHfor all g∈Gal(L/F ).
For each normal finite extension E/F with E⊆L, we have a homomorphism ψEfrom Gto the
finite group Gal(E/F ) (restrict each η∈Gto E— note that η(E) = Ebecause Eis normal). This
ψEis continuous by the definition of T(use the same E). Take the product over all Eto obtain a
homomorphism ψ=QEψEfrom Gto Γ := QEGal(E /F ). Recall that each Gal(E /F ) carries the
discrete topology; we use these to give Γ its product topology.
If E⊆E0then ψEis the composition of ψE0with the restriction map from Gal(E0/F ) to Gal(E/F ).
Thus ψ(G) is contained in the subgroup of Γ consisting of all {ηE}such that whenever E⊆E0the
image of ηE0under that restriction map is ηE. Call this group Γ0; this is the “projective limit” of
the groups Gal(E/F ) with respect to the restriction maps Gal(E0/F )→Gal(E /F ). It is a closed
subgroup of Γ, because it is the intersection of the closed subgroups obtained by imposing each
(E, E 0) condition individually.
To go further, we use the Axiom of Choice, in its familiar guise as Zorn’s Lemma. This is no great
concession because Choice is needed to even construct many of the infinite Galois extensions L/F
that interest us.
3. Show (under AC/Zorn) that ψ(G) = Γ0. Verify that ψis a homeomorphism from Gto Γ0.
