TY - JOUR
T1 - On cubic equations over p-adic fields
AU - Mukhamedov, Farrukh
AU - Omirov, Bakhrom
AU - Saburov, Mansoor
N1 - Funding Information:
The authors acknowledge the IIUM Grant EDW B13-029-0914 and the MOHE Grants ERGS13-024-0057 and ERGS13-025-0058. The authors also thank Dr Jan Kohlhaase for his attention and some comments on the paper. The first and the third authors are grateful to the Junior Associate scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The third author is also grateful to Dr Pah Chin Hee for his valued discussion and providing some references. Finally, the authors wish to express their gratitude to an anonymous referee for several helpful comments concerning a nice presentation of the paper.
PY - 2014/8
Y1 - 2014/8
N2 - We provide a solvability criterion for a cubic equation in domains Z p, Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Zp, Z p, Qp are provided. Since p is a subgroup of p, we generalize Serre's and Sun's results concerning with cubic equations over the finite field p. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.
AB - We provide a solvability criterion for a cubic equation in domains Z p, Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Zp, Z p, Qp are provided. Since p is a subgroup of p, we generalize Serre's and Sun's results concerning with cubic equations over the finite field p. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.
KW - Cubic equation
KW - p-adic Cardano formula
KW - p-adic number
KW - solvability criterion
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U2 - 10.1142/S1793042114500201
DO - 10.1142/S1793042114500201
M3 - Article
AN - SCOPUS:84904542880
SN - 1793-0421
VL - 10
SP - 1171
EP - 1190
JO - International Journal of Number Theory
JF - International Journal of Number Theory
IS - 5
ER -