Solvability criteria for the equation xq = a in the field of p-adic numbers

J. M. Casas; B. A. Omirov; U. A. Rozikov

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

J. M. Casas
, B. A. Omirov
, U. A. Rozikov

University of Vigo
Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

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).

Original language	English
Pages (from-to)	853-864
Number of pages	12
Journal	Bulletin of the Malaysian Mathematical Sciences Society
Volume	37
Issue number	3
State	Published - 2014
Externally published	Yes

Keywords

Congruence
Solvability of an equation
p-adic number

Cite this

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title = "Solvability criteria for the equation xq = a in the field of p-adic numbers",

abstract = "We establish the solvability criteria for the equation xq = 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 xq = a for any q can be reduced to the cases (i) and (ii).",

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T1 - Solvability criteria for the equation xq = a in the field of p-adic numbers

AU - Casas, J. M.

AU - Omirov, B. A.

AU - Rozikov, U. A.

PY - 2014

Y1 - 2014

N2 - We establish the solvability criteria for the equation xq = 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 xq = a for any q can be reduced to the cases (i) and (ii).

AB - We establish the solvability criteria for the equation xq = 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 xq = a for any q can be reduced to the cases (i) and (ii).

KW - Congruence

KW - Solvability of an equation

KW - p-adic number

UR - https://www.scopus.com/pages/publications/84903755370

M3 - 文章

AN - SCOPUS:84903755370

SN - 0126-6705

VL - 37

SP - 853

EP - 864

JO - Bulletin of the Malaysian Mathematical Sciences Society

JF - Bulletin of the Malaysian Mathematical Sciences Society

IS - 3

ER -

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

Abstract

Keywords

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