%0 Journal Article
%T Solvability criteria for the equation $x^q=a$ in the field of $p$-adic numbers
%A J. M. Casas
%A B. A. Omirov
%A U. A. Rozikov
%J Mathematics
%D 2011
%I arXiv
%X 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).
%U http://arxiv.org/abs/1102.2156v2