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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