2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159610We complete the program begun by Brin and Squier of characterising conjugacy in Thompson's group $F$ using the standard action of $F$ as a group of piecewise linear homeomorphisms of the unit interval.11 pages. The exposition has been rewritten. We have added an extra section which analyses the structure of roots and centralizers; the title has been changed to reflect this. The paper has been submitted for publicationGroup Theory20E45; 37E05text