The Galois structure of ambiguous ideals in cyclic extensions of degree 8

In cyclic, degree 8 extensions of algebraic number fields N/K, ambiguous ideals in N are canonical Z [C-8]-modules. Their Z[C-8]-structure is determined here. It is described in terms of indecomposable modules and determined by ramification invariants. Although infinitely many indecomposable Z[C-8]-modules are available (classification by Yakovlev), only 23 appear.