Minimal Conductors of Kummer Extensions by Roots of Unit Elements, by Romyar T. Sharifi

This paper has now appeared in J. Ramanujan Math. Soc. 16 (2001), no. 2, 101--117 and so the preprint has been removed. We consider the question of finding conductors of degree p^n Kummer extensions of a p-adic field F containing (p^n)th roots of unity. It appears to be a very difficult question to attain a reasonable recipe for these conductors in terms of those elements with (p^n)th root defining the Kummer extensions. We reduce the question of finding such a recipe to that of determining the elements with "minimal conductor" among elements a fixed distance from 1. We provide sample calculations in the case where F is the cyclotomic extension of Q_p by (p^n)th roots of 1.

Romyar T. Sharifi <sharifi@math.arizona.edu>