Let \(F\) be an algebraic function field over \({\mathbb F}_q\) and let \(P_\infty\) be a degree one place of \(F\). The author considers the group of classes of degree 0 divisors of the form \(P-P_\infty\), where \(P\) runs through the degree one places of \(F\). Let \(E\) denote the exponent of this finite Abelian group. In this note, the author obtains bounds on the number of degree one places of \(F\) in terms of \(E\) and in terms of the maximum number of degree one places of a function field of genus \((E-2)(E-1)/2\).