Solvability criteria for the equation xq=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 shown that solvability problem of x q = a for any q can be reduced to the cases (i) and (ii).