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.
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
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.[2]
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.[3][4][5]
Let K be a
finite extension of the
p-adic numbersQp. 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.[6][7] Some results are known in the case p = 2, but the structure for Q2 is not known.[8]
Another case in which the absolute Galois group has been determined is for the largest
totally real subfield of the field of algebraic numbers.[9]
Problems
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.[10]
An interesting problem is to settle
Ján Mináč and Nguyên Duy Tân's conjecture about vanishing of - Massey products for .[11][12]
Some general results
Every profinite group occurs as a Galois group of some Galois extension,[13] however not every profinite group occurs as an absolute Galois group. For example, the
Artin–Schreier theorem asserts that the only finite absolute Galois groups are either trivial or of order 2, that is only two isomorphism classes.
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,
MR1800587
Mináč, Ján; Tân, Nguyên Duy (2016), "Triple Massey products and Galois Theory", Journal of European Mathematical Society, 19 (1): 255–284
Harpaz, Yonatan; Wittenberg, Olivier (2023), "The Massey vanishing conjecture for number fields", Duke Mathematical Journal, 172 (1): 1–41
Szamuely, Tamás (2009), Galois Groups and Fundamental Groups, Cambridge studies in advanced mathematics, vol. 117,
Cambridge:
Cambridge University Press
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.
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
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.[2]
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.[3][4][5]
Let K be a
finite extension of the
p-adic numbersQp. 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.[6][7] Some results are known in the case p = 2, but the structure for Q2 is not known.[8]
Another case in which the absolute Galois group has been determined is for the largest
totally real subfield of the field of algebraic numbers.[9]
Problems
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.[10]
An interesting problem is to settle
Ján Mináč and Nguyên Duy Tân's conjecture about vanishing of - Massey products for .[11][12]
Some general results
Every profinite group occurs as a Galois group of some Galois extension,[13] however not every profinite group occurs as an absolute Galois group. For example, the
Artin–Schreier theorem asserts that the only finite absolute Galois groups are either trivial or of order 2, that is only two isomorphism classes.
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,
MR1800587
Mináč, Ján; Tân, Nguyên Duy (2016), "Triple Massey products and Galois Theory", Journal of European Mathematical Society, 19 (1): 255–284
Harpaz, Yonatan; Wittenberg, Olivier (2023), "The Massey vanishing conjecture for number fields", Duke Mathematical Journal, 172 (1): 1–41
Szamuely, Tamás (2009), Galois Groups and Fundamental Groups, Cambridge studies in advanced mathematics, vol. 117,
Cambridge:
Cambridge University Press