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

• Main Authors: Saburov, Mansoor, Ahmad, Mohd Ali Khameini
• Format: Article
• Language: English; English
• Published: Institute Mathematical Sciences, Universiti Putra Malaysia 2016
• Subjects: QA Mathematics
• Online Access: http://irep.iium.edu.my/51131/1/Bi-Quadratic_Eq_---_MJMS.pdf; http://irep.iium.edu.my/51131/4/51131-Solvability%20and%20number%20of%20roots%20of%20bi-quadratic%20equations%20over%20p-adic%20fields_SCOPUS.pdf; http://irep.iium.edu.my/51131/; http://einspem.upm.edu.my/journal/fullpaper/vol10sfeb/No2.pdf

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.