Volume 37 • Number 3 • 2014   • Solvability Criteria for the Equation $x^q=a$ in the Field of $p$-adic Numbers J. M. Casas, B. A.Omirov and U. A. Rozikov Abstract. 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). 2010 Mathematics Subject Classification: 11S05 Full text: PDF