A new lower bound for corresponding residue systems in normal, totally ramified extensions of number fields (Q1913504)

Let \(F\) be a finite extension of \(\mathbb{Q}_p\). Assume that \(K\), \(K'\) and \(L\) are finite extensions of \(F\) such that \(L\), \(K\), \(K'\) are normal over \(F\), \(KK' \subset L\) and \(K\), \(K'\) are linearly disjoint over \(F\). The author gives a lower bound for the largest integer \(m = M_L(K,K')\) satisfying \[ {\mathfrak O}_K + {\mathfrak P}^m = {\mathfrak O}_{K'} + {\mathfrak P}^m \] in the case when \(L/F\) is totally ramified, where \({\mathfrak P}\) is the (unique) maximal ideal of \({\mathfrak O}_L\). A global version of the result is also presented.