Absolute Galois Group - Examples

Examples

• 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 = 2.
• The absolute Galois group of a finite field K is isomorphic to the group

The Frobenius automorphism Fr is a canonical (topological) generator of G_K. (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.
• 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.
• Let K be a finite extension of the p-adic numbers Q_p. For p ≠ 2, its absolute Galois group is generated by + 3 elements and has an explicit description by generators and relations. This is a result of Uwe Jannsen and Kay Wingberg. Some results are known in the case p = 2, but the structure for Q₂ is not known.
• Another case in which the absolute Galois group has been determined is for the largest totally real subfield of the field of algebraic numbers.