Carlitz Modules and Galois Module Structure II Original Research Article

LetLsuperset ofFbe cyclotomic function fields of Carlitz. We show that ifL/Fis Carlitz–Kummer, the integer ringOLis free over the associated order as in the classical cyclotomic Kummer extension. However, contrary to the characteristic zero case,OLis not free unlessL/Fis Carlitz–Kummer.