The absolute Galois group of C(x)

Dan Haran and Moshe Jarden

Vol. 196 (2000), No. 2, 445–459

DOI: 10.2140/pjm.2000.196.445

Abstract

We use elementary algebraic methods to reprove a theorem which was proved by Pop using rigid analytic geometry and in a less general form by Harbater using formal algebraic patching:

Let C be an algebraically closed field of cardinality m. Consider a subset S of ℙ¹(C) of cardinality m. Then the fundamental group of ℙ¹(C)\ S is isomorphic to the free profinite group of rank m.

We also observe that if char(C)≠0 and 0 < card(S) < m, then π₁(ℙ¹(C)\ S) is not isomorphic to a free profinite group.

Milestones

Received: 28 April 1999

Revised: 19 November 1999

Published: 1 December 2000