Solvability and number of roots of bi-quadratic equations over p−adic fields

Unlike the real number field R, a bi-quadratic equation x4 + 1 = 0 is solvable over some p−adic number fields Qp, 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 x4 + ax2 = b over domains Z ∗ p, Zp \ Z ∗ p, Qp \ Zp, Qp, where p > 2. Moreover, we also provide the number of roots of the bi-quadratic equation over the mentioned domains