Solvability criteria for the equation $x^q=a$ in the field of $p$-adic numbers

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 showed that solvability problem of $x^q=a$ for any $q$ can be reduced to the cases (i) and (ii).

14 pages

References in corpus (1)

• On equation $x^q=a$ over $\bq_p$

Cited by in corpus (3)

• On equation $x^q=a$ over $\bq_p$
• On cubic equations over $P-$adic field
• Classification of p-adic 6-dimensional filiform Leibniz algebras by solution of x^q=a