Let K be a. complete non-archimedean field with a discrete valuation, f K[X] a polynomial with non-vanishing discriminant, A the valuation ring of K, and m the maximal ideal of A. The first main result of this paper is a reformulation of Hensel's lemma that connects the number of roots of f with the number of roots of its reduction modulo a power of m We then define a condition -regularity -that yields a simple method to compute the exact number of roots of f in K. In particular, we show that regularity implies that the number of roots of f equals the sum of the numbers of roots of certain binomials derived from the Newton polygon. 2010 University of Houston.
