On the Galois module structure of ideals and rings of all integers of \({\mathfrak p}\)-adic number fields (Q1904230)

For a finite Galois extension \(K/k\) of local number fields with residue field of characteristic \(p\) a complete description of ideals in \(o_K\) as \(o_k [G]\)-modules, \(G= G(K/ k)\), is still unknown when the order of \(G\) is divisible by \(p\). The simplest cases studied earlier with partial success for \(G\) being a \(p\)-group, \(p>2\), are: (1) \(K/k\) is a cyclic Kummer extension, (2) the ramification group \(G_1\) of \(G\) is not cyclic. Applying results of \textit{S. Vostokov} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklov (LOMI) 46, 14-35 (1974; Zbl 0345.12009); ibid. 57, 64-84 (1976; Zbl 0355.12012)]\ and himself [Ill. J. Math. 31, 185-199 (1987; Zbl 0611.12013)]\ the author lists several new observations in these two cases. In particular, he shows that one can reduce the study of ideals in \(o_K\) in the first case (\(o_K\) in the second case) as \(o_k [G]\)-modules to the study of \(o_k [G_1]\)- modules.