On the Solutions of the Equation *x*^{3} + *Ax* = *B* in Z_{3}^{*} with Coefficients from Q_{3}

Affiliation(s)

Universiti Kuala Lumpur, Malaysian Institute of Industrial Technology, Johor Bahru, Malaysia.

Institute of Mathematics, Tashkent, Uzbekistan.

Universiti Kuala Lumpur, Malaysian Institute of Industrial Technology, Johor Bahru, Malaysia.

Institute of Mathematics, Tashkent, Uzbekistan.

ABSTRACT

Recall
that in [1] it is obtained the criteria solvability of the Equation in , and for *P*>3. Since any *p*-adic number *x* has a unique form , where and in [1] it is also shown that from the criteria
in it follows the criteria in and . In this
paper we provide the algorithm of finding the solutions of the Equation in with coefficients from .

Cite this paper

I. Rikhsiboev, A. Khudoyberdiyev, T. Kurbanbaev and K. Masutova, "On the Solutions of the Equation*x*^{3} + *Ax* = *B* in Z_{3}^{*} with Coefficients from Q_{3}," *Applied Mathematics*, Vol. 5 No. 1, 2014, pp. 35-46. doi: 10.4236/am.2014.51005.

I. Rikhsiboev, A. Khudoyberdiyev, T. Kurbanbaev and K. Masutova, "On the Solutions of the Equation

References

[1] F. M. Mukhamedov, B. A. Omirov, M. Kh. Saburov and K. K. Masutova, “Solvability of Cubic Equations in p-Adic Integers ,” Siberian Mathematical Journal, Vol. 54, No. 3, 2013, pp. 501-516.

[2] K. Hensel, “Untersuchung der Fundamentalgleichung Einer Gattung fur Eine Reelle Primzahl als Modul und Besrimmung der Theiler Ihrer Discriminante,” Journal Für Die Reine und Angewandte Mathematik, Vol. 113, No. 1, 1894, pp. 61-83.

[3] A. A. Buhshtab, “Theory of Numbers,” Moscow, 1966, 384 p.

[4] S. B. Katok, “p-Adic Analysis Compared with Real,” MASS Selecta 2004.

[5] N. Koblitz “p-Adic Numbers, p-Adic Analysis and Zeta-Functions,” Springer-Verlag, New York, Heidelberg, Berlin, 1977, 190 p. http://dx.doi.org/10.1007/978-1-4684-0047-2

[6] V. S. Vladimirov, I. V. Volovich and I. Zelenov, “p-Adic Analysis and Mathematical Physics” World Scientific, Singapore City, 1994. http://dx.doi.org/10.1142/1581

[7] S. Albeverio, Sh. A. Ayupov, B. A. Omirov and A. Kh. Khudoyberdiyev, “n-Dimensional Filiform Leibniz Algebras of Length (n-1) and Their Derivations,” Journal of Algebra, Vol. 319, No. 6, 2008, pp. 2471-2488.

http://dx.doi.org/10.1016/j.jalgebra.2007.12.014

[8] Sh. A. Ayupov and B. A. Omirov, “On Some Classes of Nilpotent Leibniz Algebras,” Siberian Mathematical Journal, Vol. 42, No. 1, 2001, pp. 18-29. http://dx.doi.org/10.1023/A:1004829123402

[9] B. A. Omirov and I. S. Rakhimov, “On Lie-Like Complex Filiform Leibniz Algebras,” Bulletin of the Australian Mathematical Society, Vol. 79, No. 3, 2009, pp. 391-404.

http://dx.doi.org/10.1017/S000497270900001X

[10] I. S. Rakhimov and S. K. Said Husain, “On Isomorphism Classes and Invariants of Low Dimensional Complex Filiform Leibniz Algebras,” Linear and Multilinear Algebra, Vol. 59, No. 2, 2011, pp. 205-220.

http://dx.doi.org/10.1080/03081080903357646

[11] Sh. A. Ayupov and T. K. Kurbanbaev, “The Classification of 4-Dimensional p-Adic Filiform Leibniz Algebras,” TWMS Journal of Pure and Applied Mathematics, Vol. 1, No. 2, 2010, pp. 155-162.

[12] A. Kh. Khudoyberdiyev, T. K. Kurbanbaev and B. A. Omirov, “Classification of Three-Dimensional Solvable p-Adic Leibniz Algebras,” p-Adic Numbers, Ultrametric Analysis and Applications, Vol. 2, No. 3, 2010, pp. 207-221.

[13] M. Ladra, B. A. Omirov and U. A. Rozikov, “Classification of p-Adic 6-Dimensional Filiform Leibniz Algebras by Solution of ,” Central European Journal of Mathematics, Vol. 11, No. 6, 2013, pp. 1083-1093.

http://dx.doi.org/10.2478/s11533-013-0225-9

[14] J. M. Casas, B. A. Omirov and U. A. Rozikov, “Solvability Criteria for the Equation in the Field of p-Adic Numbers,” 2011. arXiv:1102.2156v1

[15] F. M. Mukhamedov and M. Kh. Saburov, “On Equation over ,” Journal of Number Theory, Vol. 133, No. 1, 2013, pp. 55-58.

[16] T. K. Kurbanbaev and K. K. Masutova, “On the Solvability Criterion of the Equation in with Coefficients from ,” Uzbek Mathematical Journal, No. 4, 2011, pp. 96-103.

[1] F. M. Mukhamedov, B. A. Omirov, M. Kh. Saburov and K. K. Masutova, “Solvability of Cubic Equations in p-Adic Integers ,” Siberian Mathematical Journal, Vol. 54, No. 3, 2013, pp. 501-516.

[2] K. Hensel, “Untersuchung der Fundamentalgleichung Einer Gattung fur Eine Reelle Primzahl als Modul und Besrimmung der Theiler Ihrer Discriminante,” Journal Für Die Reine und Angewandte Mathematik, Vol. 113, No. 1, 1894, pp. 61-83.

[3] A. A. Buhshtab, “Theory of Numbers,” Moscow, 1966, 384 p.

[4] S. B. Katok, “p-Adic Analysis Compared with Real,” MASS Selecta 2004.

[5] N. Koblitz “p-Adic Numbers, p-Adic Analysis and Zeta-Functions,” Springer-Verlag, New York, Heidelberg, Berlin, 1977, 190 p. http://dx.doi.org/10.1007/978-1-4684-0047-2

[6] V. S. Vladimirov, I. V. Volovich and I. Zelenov, “p-Adic Analysis and Mathematical Physics” World Scientific, Singapore City, 1994. http://dx.doi.org/10.1142/1581

[7] S. Albeverio, Sh. A. Ayupov, B. A. Omirov and A. Kh. Khudoyberdiyev, “n-Dimensional Filiform Leibniz Algebras of Length (n-1) and Their Derivations,” Journal of Algebra, Vol. 319, No. 6, 2008, pp. 2471-2488.

http://dx.doi.org/10.1016/j.jalgebra.2007.12.014

[8] Sh. A. Ayupov and B. A. Omirov, “On Some Classes of Nilpotent Leibniz Algebras,” Siberian Mathematical Journal, Vol. 42, No. 1, 2001, pp. 18-29. http://dx.doi.org/10.1023/A:1004829123402

[9] B. A. Omirov and I. S. Rakhimov, “On Lie-Like Complex Filiform Leibniz Algebras,” Bulletin of the Australian Mathematical Society, Vol. 79, No. 3, 2009, pp. 391-404.

http://dx.doi.org/10.1017/S000497270900001X

[10] I. S. Rakhimov and S. K. Said Husain, “On Isomorphism Classes and Invariants of Low Dimensional Complex Filiform Leibniz Algebras,” Linear and Multilinear Algebra, Vol. 59, No. 2, 2011, pp. 205-220.

http://dx.doi.org/10.1080/03081080903357646

[11] Sh. A. Ayupov and T. K. Kurbanbaev, “The Classification of 4-Dimensional p-Adic Filiform Leibniz Algebras,” TWMS Journal of Pure and Applied Mathematics, Vol. 1, No. 2, 2010, pp. 155-162.

[12] A. Kh. Khudoyberdiyev, T. K. Kurbanbaev and B. A. Omirov, “Classification of Three-Dimensional Solvable p-Adic Leibniz Algebras,” p-Adic Numbers, Ultrametric Analysis and Applications, Vol. 2, No. 3, 2010, pp. 207-221.

[13] M. Ladra, B. A. Omirov and U. A. Rozikov, “Classification of p-Adic 6-Dimensional Filiform Leibniz Algebras by Solution of ,” Central European Journal of Mathematics, Vol. 11, No. 6, 2013, pp. 1083-1093.

http://dx.doi.org/10.2478/s11533-013-0225-9

[14] J. M. Casas, B. A. Omirov and U. A. Rozikov, “Solvability Criteria for the Equation in the Field of p-Adic Numbers,” 2011. arXiv:1102.2156v1

[15] F. M. Mukhamedov and M. Kh. Saburov, “On Equation over ,” Journal of Number Theory, Vol. 133, No. 1, 2013, pp. 55-58.

[16] T. K. Kurbanbaev and K. K. Masutova, “On the Solvability Criterion of the Equation in with Coefficients from ,” Uzbek Mathematical Journal, No. 4, 2011, pp. 96-103.