Geodesic
Solvability Criteria for the Equation $x^q=a$ in the Field of $p$-adic Numbers
Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 3 Cet article a éte moissonné depuis la source Bulletin of the Malaysian Mathematical Society website

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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).
Classification : 11S05 
@article{BMMS_2014_37_3_a20,
     author = {J. M. Casas and B. A.Omirov and U. A. Rozikov},
     title = {Solvability {Criteria} for the {Equation} $x^q=a$ in the {Field} of $p$-adic {Numbers}},
     journal = {Bulletin of the Malaysian Mathematical Society},
     year = {2014},
     volume = {37},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/BMMS_2014_37_3_a20/}
}
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J. M. Casas; B. A.Omirov; U. A. Rozikov. Solvability Criteria for the Equation $x^q=a$ in the Field of $p$-adic Numbers. Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 3. http://geodesic.mathdoc.fr/item/BMMS_2014_37_3_a20/