The absolute galois group of C(x)

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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.

Original languageEnglish
Pages (from-to)445-459
Number of pages15
JournalPacific Journal of Mathematics
Volume196
Issue number2
DOIs
StatePublished - Dec 2000

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