2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/120384A proof of freeness of the commutator subgroup of the fundamental group of a smooth irreducible affine curve over a countable algebraically closed field of nonzero characteristic. A description of the abelianizations of the fundamental groups of smooth affine curves over an algebracially closed field of nonzero characteristic is also given.The main theorem (Theorem 4.1) now asserts the result for only countable fields. Proofs of patching results have been simplifiedAlgebraic GeometryCommutative Algebra12E30; 14H30text