I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book:
Let E/K be a finite Galois extension of a complete field K. Prolong the valuation of K to E. Let x be in E and let {x_1, x_2,...,x_n} be the set of conjugates of x over K, with x=x_1. Let y in E be such that ||y-x|| less than ||y-x_i|| for all i greater or equal to 2. Show that x belongs to the field K(y).
(Note that if x_i is conjugate to x over K(y) then ||y-x||=||y-x_i|| )