Decomposition of ideals in abelian \(p\)-extensions of complete discrete valuation fields
Decomposition of ideals as Galois modules is studied here. Let \(k\) be a local field of characteristic zero, complete with a discrete valuation, having a residue field of characteristic \(p > 2\). Let \(K/k\) be a totally ramified cyclic extension of degree \(p^m\). The authors find a decomposition for any ideal \(A\) of \(K\) as an \(O[G]\)-module, where \(O\) is the ring of integers of \(k,G=\text{Gal}(K/k)\). Earlier, they gave conditions for an ideal \(A\) being decomposable as \(O[G]\)-module in [Decomposability of ideals as Galois modules in complete discrete valuation fields, St. Petersbg. Math J. 11, No. 2, 233-249 (2000); translation from Algebra Anal. 11, No. 2, 41-63 (1999; Zbl 0964.11054)]. The more general case where \(K/k\) is an abelian \(p\)-extension could be deduced to the above case.
scientific article; zbMATH DE number 1337563