Über Arithmetische Assoziierte Ordnungen Original Research Article

Letpbe an odd rational prime, andL/Ka totally ramified finite abelian extension ofp-adic fields. LetGdenote the Galois group ofL/K. For each idealIof the valuation ring ofLwe study its “associated order”Endimagep[G](I) with respect to imagep[G]. This order is a natural operating ring for the idealI, and is an important invariant of its imagep[G]-structure. We show that unlessGis cyclic no such order can contain any idempotent of imagep[G] which does not already lie in imagep[G] This is a strong restriction on the possible structures of arithmetical associated orders. We also describe an algorithm for the explicit computation of associated orders which can in many cases be implemented effectively on a computer. In this way we are able to compute explicit examples of arithmetical associated orders which falsify a conjecture previously made by the second named author.