I will post my solutions to a few problems i've considered here, I just need feedback from people who can tell me if my ideas are correct because i'm feeling shaky about the methods. Z7 is the dvr in the completion of the rationals w.r.t. the 7adic metric. Then for what integers a is there a solution to the polynomial 7x^2-a in Z7. If there was one then root(a/7) is in Z7 so it's square a/7 is too, thus |a/7| can be no bigger than one i.e. 7 divides a in the integers. We are looking at solutions of x^2-a/7 and these exist if and only if solutions exist modulo 7 (necessity is obvious and Hensel's lemma gives sufficiency). The integers a for which a/7 is a quadratic residue (mod7) are exactly those that are congruent to 0,7,14, or 28 modulo 49. Now I'm wondering about the same question for what rational 7adic numbers (aka Q7), a, does there exist a solution to the equation in Q7? I will post about this when I have an idea.