Solvability and Number of Roots of Bi-Quadratic Equations over p-adic Fields

Unlike the real number field R, a bi-quadratic equation x(4) + 1 = 0 is solvable over some p-adic number fields 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 Qp. In this paper, we provide solvability criteria for the bi-quadratic equation x(4) + ax 2 = b over domains Z*(p), Z(p) \ Z*(p), Q(p) \ Z(p), Q(p), where p > 2. Moreover, we also provide the number of roots of the bi-quadratic equation over the mentioned domains.