Let p >= 3 be a prime number. In this article, we study the canonical expansion of the primitive pn-th root of unity zeta(n)(p) in p-adicMal'cev-Neumann field L-p for n >= 1. More precisely, we give the explicit formula for the first aleph(0) terms of the expansion of.pn, and as an application, we use it to construct a uniformizer of K-2,K-m = Q(p) (zeta(2)(p), p(1/pm)) with m >= 1.
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