On automorphisms of order three of division algebras (Q1109861)

The authors consider finite dimensional (not necessarily associative) division algebras A over a field F. Suppose G is a cyclic subgroup in Aut A of order \(p^ n\), where p is a prime and n a positive integer. Fix a generator g in G. If a minimal polynomial of g splits over F into linear factors, then A is a free FG-module. This is always the case if char F\(=p.\) In particular, let \(p=3\) and \(n=1\). Then in the modular case by the preceding result A is a free FG-module. In the non-modular case A is a direct sum of irreducible FG-modules. Examples are given to show that any combination of irreducible FG-modules does really exist.