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| Gal(Qbar/Q) | |
|---|---|
| Name | Absolute Galois group of the rationals |
| Notation | Gal(Qbar/Q) |
| Type | Profinite group |
| Related | Galois theory, Algebraic number theory, Étale cohomology |
Gal(Qbar/Q) is the absolute Galois group of the field of rational numbers, the group of field automorphisms of an algebraic closure that fix the rational numbers. It plays a central role in Galois theory, Algebraic number theory, and the formulation of reciprocity laws in Class field theory, connecting arithmetic of Q with representations in Group theory, Algebraic geometry, and Topology.
Gal(Qbar/Q) is defined as the group of continuous automorphisms of an algebraic closure Qbar of Q that fix Q pointwise; it is a profinite group, inverse limit of finite Galois groups Gal(K/Q) as K ranges over finite Galois extensions. Basic invariants include its profinite topology, its decomposition into projective limits of Galois groups of number fields like Cyclotomic fields, and the Krull topology coming from finite quotients such as Gal(Q(ζ_n)/Q). Important classical facts link it to Frobenius automorphisms for primes, the Chebotarev density theorem, and the description of abelianizations via Kronecker–Weber theorem and Class field theory.
As a profinite group, Gal(Qbar/Q) is compact, totally disconnected, and Hausdorff; it is presented as lim← Gal(K/Q) for finite Galois K. Its maximal pro-p quotients, prosolvable quotients, and maximal abelian quotient Gal(Qbar/Q)^{ab} are studied via Iwasawa theory, Local field analogues such as Gal(Q_p^bar/Q_p), and global-to-local maps given by embedding decomposition groups at primes like those of p-adic numbers and Archimedean place. Deep conjectures such as the Fontaine–Mazur conjecture and Grothendieck's anabelian conjecture constrain possible topological and algebraic structures. Representations into groups like GL_n(F_p), GL_n(Z_p), and GL_n(C) appear in the theory of Galois representations, Modular forms, and Langlands program correspondences.
For each prime ideal of a number field lying over a rational prime, Gal(Qbar/Q) contains decomposition and inertia subgroups isomorphic to subgroups of local Galois groups such as Gal(Q_p^bar/Q_p). Decomposition groups house Frobenius elements corresponding to prime numbers via Frobenius conjugacy classes, and inertia groups control ramification phenomena classified by Wild ramification and tamely ramified extensions described by Kummer theory and Cyclotomic extensions. Local-global compatibility in the behavior of inertia is reflected in reciprocity maps from Idèle class groups and in compatibilities with Hodge–Tate decomposition for p-adic representations.
Gal(Qbar/Q) acts naturally on algebraic varieties defined over Q, points of schemes such as Spec(Z[1/N]) and on étale fundamental groups of curves like P^1 minus points. This action underlies the formulation of the Étale cohomology of varieties, the action on torsion points of Elliptic curves, and the study of motivic Galois groups and Tate modules. Key links include the action on the pro-étale fundamental group featured in Grothendieck–Teichmüller theory and interactions with objects such as Moduli spacees of curves and the Mumford–Tate group in Hodge theory.
Finite quotients of Gal(Qbar/Q) are precisely finite Galois groups over Q, so the study of realizable finite groups over Q is the classical Inverse Galois problem. Known realizations involve groups like cyclic groups from Cyclotomic fields, symmetric groups S_n, alternating groups A_n, and many finite simple groups achieved via techniques from Hilbert irreducibility theorem, Rigidity method, and Shafarevich theorem for solvable groups. Deep instances connect to the realization of groups of Lie type, use of Modular curves, Galois representations attached to Modular forms, and explicit constructions via specialization of covers.
Cohomology groups H^i(Gal(Qbar/Q), M) for discrete modules M classify extensions, torsors, and Brauer groups; they feature in duality theorems such as Tate duality and in arithmetic duality linking Selmer groups and Sha. The cohomological dimension, Massey products, and pro-p cohomology detect obstructions described in Bloch–Kato conjecture (now a theorem via Voevodsky and collaborators) and constrain realizable Galois extensions. Profinite group invariants include Frattini subgroups, Demuškin groups in local contexts, and presentation ranks studied in Golod–Shafarevich theorem applications to infinite unramified extensions.
Known subgroups arise as decomposition groups at primes, inertia subgroups, and open subgroups corresponding to absolute Galois groups of number fields like Q(√d), Imaginary quadratic fields, and Cyclotomic extensions. Classification results include descriptions of maximal abelian quotient via Kronecker–Weber theorem, prosolvable realizations via Shafarevich theorem, and structural constraints from Neukirch–Uchida theorem identifying number fields from their absolute Galois groups. Open problems remain: full characterization of all closed subgroups, absolute determination of all finite quotients (Inverse Galois problem), and the structural consequences of conjectures in the Langlands program and Anabelian geometry.
Category:Galois groups Category:Algebraic number theory