2 SARA ARIAS-DE-REYNA
Problem 1.2 (Inverse Galois Problem). Let G be a finite group. Does there
exist a Galois extension L/Q such that Gal(L/Q) G?
The first mathematician that addressed this problem was D. Hilbert. In his
paper [13] he proves his famous Irreducibility Theorem, and applies it to show that,
for all n N, the symmetric group Sn and the alternating group An occur as Galois
groups over the rationals. Since then, many mathematicians have thought about
the inverse Galois problem, and in fact it is now solved (affirmatively) for many
(families of) finite groups G. For instance, let us mention the result of Shafarevich
that all solvable groups occur as Galois groups over the rationals (see [23] for a
detailed explanation of the proof). However, it is still not known if the answer is
affirmative for every finite group G, and as far as I know, there is no general strategy
that addresses all finite groups at once. An account of the different techniques used
to address the problem can be found in [32].
Let K be a field, and let us fix a separable closure Ksep. There is a way to
group together all the Galois groups of finite Galois extensions L/K contained in
Ksep, namely the absolute Galois group of K. It is defined as the inverse limit
GK := Gal(Ksep/K) = lim
←−
L/K
finite Galois
Gal(L/K).
This group is a profinite group, and as such is endowed with a topology, called
the Krull topology, which makes it a Hausdorff, compact and totally disconnected
group. A very natural question to ask is what information on the field K is encoded
in the topological group GK. In this connection, a celebrated result of Neukirch,
Iwasawa, Uchida and Ikeda establishes that, if K1,K2 are two finite extensions of
Q contained in a fixed algebraic closure Q such that GK1 GK2 , then K1 and K2
are conjugated by some element in GQ (cf. [34], [16]). Let us note, however, that
we cannot replace Q by any field. For example, the analogous statement does not
hold when the base field is Qp, cf. [37] and [17]. Thus, we see that the absolute
Galois group of Q encodes a wealth of information about the arithmetic of number
fields. In this context, the inverse Galois problem can be formulated as the question
of determining which finite groups occur as quotient groups of GQ.
A natural way to study GQ is to consider its representations, that it, the con-
tinuous group morphisms GQ GLm(k), where k is a topological field and m N.
Such a representation will be called a Galois representation. Let us assume that k
is a finite field, endowed with the discrete topology, and let
ρ : GQ GLm(k)
be a Galois representation. Since the set {Id} is open in GLm(k), we obtain that
ker ρ GQ is an open subgroup. In other words, there exists a finite Galois
extension K/Q such that ker ρ = GK . Therefore
Imρ GQ/ ker ρ GQ/GK Gal(K/Q).
This reasoning shows that, whenever we are given a Galois representation of
GQ over a finite field k, we obtain a realisation of Imρ GLm(k) as a Galois group
over Q. In this way, any source of Galois representations provides us with a strategy
to address the inverse Galois problem for the subgroups of GLm(k) that occur as
images thereof.
Geometry provides us with many objects endowed with an action of the abso-
lute Galois group of the rationals, thus giving rise to such Galois representations.
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