Absolute Galois group

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The absolute Galois group of the real numbers is a cyclic group of order 2 generated by complex conjugation, since C is the separable closure of R and [C:R] = 2.

In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K; the absolute Galois group is well-defined up to inner automorphism. It is a profinite group.

(When K is a perfect field, Ksep is the same as an algebraic closure Kalg of K. This holds e.g. for K of characteristic zero, or K a finite field.)

Examples[edit]

  • The absolute Galois group of an algebraically closed field is trivial.
  • The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R and [C:R] = 2.
  • The absolute Galois group of a finite field K is isomorphic to the group

(For the notation, see Inverse limit.)

The Frobenius automorphism Fr is a canonical (topological) generator of GK. (Recall that Fr(x) = xq for all x in Kalg, where q is the number of elements in K.)
  • The absolute Galois group of the field of rational functions with complex coefficients is free (as a profinite group). This result is due to Adrien Douady and has its origins in Riemann's existence theorem.[1]
  • More generally, let C be an algebraically closed field and x a variable. Then the absolute Galois group of K = C(x) is free of rank equal to the cardinality of C; this result is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden using algebraic methods.[2][3][4]
  • Let K be a finite extension of the p-adic numbers Qp. For p ≠ 2, its absolute Galois group is generated by [K:Qp] + 3 elements and has an explicit description by generators and relations. This is a result of Uwe Jannsen and Kay Wingberg;[5][6] some results are known in the case p = 2, but the structure for Q2 is not known.[7]
  • Another case in which the absolute Galois group has been determined is for the largest totally real subfield of the field of algebraic numbers.[8]

Problems[edit]

  • No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the absolute Galois group has a faithful action on the dessins d'enfants of Grothendieck (maps on surfaces), enabling us to "see" the Galois theory of algebraic number fields.
  • Let K be the maximal abelian extension of the rational numbers. Then Shafarevich's conjecture asserts that the absolute Galois group of K is a free profinite group.[9]

Some general results[edit]

Notes[edit]

References[edit]

  • Douady, Adrien (1964), "Détermination d'un groupe de Galois", Comptes Rendus de l'Académie des Sciences de Paris, 258: 5305–5308, MR 0162796
  • Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 11 (3rd ed.), Springer-Verlag, ISBN 978-3-540-77269-9, Zbl 1145.12001
  • Haran, Dan; Jarden, Moshe (2000), "The absolute Galois group of C(x)", Pacific Journal of Mathematics, 196 (2): 445–459, doi:10.2140/pjm.2000.196.445, MR 1800587
  • Harbater, David, "Fundamental groups and embedding problems in characteristic p", Recent developments in the inverse Galois problem, Contemporary Mathematics, 186, Providence, RI: American Mathematical Society, pp. 353–369, MR 1352282
  • Jannsen, Uwe; Wingberg, Kay (1982), "Die Struktur der absoluten Galoisgruppe -adischer Zahlkörper", Inventiones Mathematicae, 70: 71–78, Bibcode:1982InMat..70...71J, doi:10.1007/bf01393199
  • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001
  • Pop, Florian (1995), "Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture", Inventiones Mathematicae, 120 (3): 555–578, Bibcode:1995InMat.120..555P, doi:10.1007/bf01241142, MR 1334484