On saturated distinguished chains over a local field II

• Published in: Journal of number theory 2005-03, Vol.111 (1), p.86-143
• Main Author: Ota, Kaori
• Format: Article
• Language: English
• Subjects: Local field; Ramification theory; Saturated distinguished chain; Wild ramification
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/j.jnt.2004.12.005

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 α 1 in a SDC for α = π 1 + π 1 π considered in (Ota, 1999) for wildly totally ramified Galois extensions, and this leads us to consider a tower of fields, K ⊂ K 1 ⊂ K 2 ⊂ … , where K 1 = K ( α 1 ) and K n / K is wildly totally ramified. The union of these fields K = ⋃ n = 1 ∞ K n 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 = π 1 + π 1 π 2 + ⋯ + π 1 π 2 ⋯ π n for totally ramified Galois extensions of type ( m , m ,..., m ) , where m is a power of the characteristic of the residual field of K.