The modular group PSL 2 ℤ contains many subgroups that are sharply transitive on the rational projective line ℚ ∪ ∞. If the field E is a purely transcendental extension of transcendency degree | E|, then PSL 2 E contains free subgroups that are sharply transitive on the projective line E ∪ ∞. This yields wild infinite nearfields and sharply two-transitive permutation groups.