The fundamental group of affine curves in positive characteristic (Q397859)

Let \(C\) be a smooth affine curve over a field \(k\) of positive characteristic. Denote by \(\pi_1^c (C)\) be the commutator subgroup of the algebraic fundamental group of \(C\). The author shows that \(\pi_1^c (C)\) is a free profinite group of rank the cardinality of \(k\), a result previously known only when \(k\) was countable.