In 1988, S. White proved by means of field theory supplemented by a geometric argument that the
real bijections
x
x+1 and
x
xd(
dan odd prime) generate a
free group of rank 2. When these maps are considered in prime characteristic
p(so that
x
x+1 generates a cyclic group of order
p) the geometric argument is no longer available. We show on the one hand that, generally, the
geometry is redundant and on the other that, in characteristic
p, further
algebraic considerations are required to establish a key polynomial lemma. By these means we obtain an analogue of White's theorem for certain (countably) infinite subfields
Lof the
algebraic closure of the finite prime field GF(
p). For any (odd) prime
d, not a divisor of
p(
p−1), the maps
x
x+1 and
x
xdgenerate a group of
bijectionsof such a field
Lthat is isomorphic to the
free product Z*(Z/
pZ). This implies an explicit natural
algebraic faithful representation of the
free product as a transitive permutation group on a countable set.