On saturated distinguished chains over a local field II

This is a continuation of the paper (J Number Theory 79 (1999) 217) on saturated distinguished chains (SDCs) over a local field K. Here, we give a “canonical” choice of the next element α1 in a SDC for α=π1+π1π considered in (Ota, 1999) for wildly totally ramified Galois extensions, and this leads us to consider a tower of fields, K⊂K1⊂K2⊂…, where K1=K(α1) and Kn/K is wildly totally ramified. The union of these fields K=⋃n=1∞Kn is particularly interesting, for its conductor over K is very small, close to 1. Moreover, in some cases K is uniquely determined up to isomorphism over K for any such extensions of the same type. We also consider SDCs for an element α=π1+π1π2+⋯+π1π2⋯πn for totally ramified Galois extensions of type (m,m,...,m), where m is a power of the characteristic of the residual field of K.