Indecomposability of ideals of \(\mathfrak p\)-adic number fields (Q1827517)

Let \(p\) be an odd prime and let \(K/k\) be a \(p\)-extension of local fields with Galois group \(G\). The author proves that any ideal \(A\) of \(K\) is \(O[G]\)-indecomposable if and only if \(| G_1|\) does not divide the different \(D_{K/k}\). Here \(O\) denotes the ring of integers of \(k\) and \(G_1\) denotes the first ramification group of \(K/k\). \textit{S. V. Vostokov} [Vestn. St. Petersbg. Univ., Math. 26, No. 2, 10--21 (1993; Zbl 0841.11059)] proved the same result in the special case that \(K\) is the composite of an unramified extension with a fully ramified extension of \(k\).