Decomposability of ideals in splitting \(p\)-extensions of local fields (Q1901841)

Let \(k\) be a \(p\)-adic number field. A finite extension \(K\) of \(k\) is called splitting if it is the composite of an unramified and a fully ramified subextension. Let \(K/k\) be a normal extension and let \(O\) be the ring of integral elements in \(k\). An ideal of \(K\) is called decomposable if it is decomposable as \(O[G(K/ k)]\)-module. The main result of the author is the following theorem: Let \(K/k\) be a normal splitting \(p\)-extension and let \(p\neq 2\). Then \(K\) contains decomposable ideals if and only if the ramification index of \(K/k\) divides the different of \(K/k\).