Malaysian Journal of Mathematical Sciences, February 2016, Vol. 10(S)
Special Issue: The 3rd International Conference on Mathematical Applications in Engineering 2014 (ICMAE'14)
Solvability and Number of Roots of Bi-Quadratic Equations over $p$-adic Fields
Saburov, M. and Ahmad, M. A. K.
Corresponding Email: [email protected]
Received date: -
Accepted date: -
Abstract:
Unlike the real number field $\mathbb{R}$, a bi-quadratic equation $x^{4}+1=0$ is solvable over some $p$-adic number fields $\mathbb{Q}_{p}$, say $p=17, 41, ...$. Therefore, it is of independent interest to provide a solvability criterion for the bi-quadratic equation over $p$-adic number fields $\mathbb{Q}_{p}$. In this paper, we provide solvability criteria for the bi-quadratic equation $x^{4}+ax^{2}=b$
over domains $\mathbb{Z}_{p}^{*}$, $\mathbb{Z}_{p}$ \ $\mathbb{Z}_{p}^{*}$, $\mathbb{Q}_{p}$ \ $\mathbb{Z}_{p}$, $\mathbb{Q}_{p}$, where $p > 2$. Moreover, we also provide the number of roots of the bi-quadratic equation over the mentioned domains.
Keywords: Bi-quadratic equation, $p$-adic number, solvability criterion
