Solvability criteria for the equation $x^q=a$ in the field of $p$-adic   numbers
  We establish the solvability criteria for the equation $x^q=a$ in the field of $p$-adic numbers, for any $q$ in two cases:   (i) $q$ is not divisible by $p$; (ii) $q=p$. Using these criteria we show that any $p$-adic number can be represented in finitely many different forms and we describe the algorithms to obtain the corresponding representations. Moreover it is showed that solvability problem of $x^q=a$ for any $q$ can be reduced to the cases (i) and (ii). 
J.M. Casas, B.A.Omirov, U.A. Rozikov
math.RA math.NT