COMPOSITES OF TRANSLATIONS AND ODD RATIONAL POWERS ACT FREELY

It is shown that no non-trivial composition of translations x --> x + a and odd rational powers x -->(p/q), where p,q are odd co-prime integers, positive or negative with p/q not equal +/-, acts like the identity on a held of characteristic zero. This extends a theorem of Adeleke, Glass, and Morley in which only odd positive rational powers were considered. Moreover, the nature of the proof itself (by field theory) is a simplification and natural refinement of previous proofs. It has applications in other settings.