Solvability of Cubic Equations over Q3

Abstract

We provide a solvability criterion for a cubic equation in domains ,  3 , and  3 .

The Number of Solutions of Cubic Equations over Q3

“…The proof is the similar to the proof in the case  3 presented in Saburov and Ahmad (2015). However, for reasons of self-exposition, we present the solvability criterion in the domains ,  3 \ ,  3 \  3 and  3 .…”

“…Case I.1: |a| 3 = |b| 3 > 1. In this case, we showed (Saburov & Ahmad 2015) that the number of solutions of the equation (4) . In this case, we showed (Saburov & Ahmad 2015) that the number of solutions of the equation (4) .…”

“…In this case, we showed (Saburov & Ahmad 2015) that the number of solutions of the equation (4) . In this case, we showed (Saburov & Ahmad 2015) that the number of solutions of the equation (4) . In this case, we showed (Saburov & Ahmad 2015) that the number of solutions of the equation (4) (4) in is the same as the number of solutions of the equation (4) in is the same as the number of solutions of the equation x 3 + ax -b ≡ 0 (mod 243).…”

“…over the domains  3 \ and  3 \  3 as a complementary part of Saburov and Ahmad (2015) paper, where a, b ∈  3 and ab ≠ 0. The proof is the similar to the proof in the case  3 presented in Saburov and Ahmad (2015).…”

“…Therefore, finding a solvability criterion for the depressed cubic equation (3) in domains ,  3 ,  3 \ ,  3 \  3 and  3 is of independent interest. In Saburov and Ahmad (2015), the solvability criterion for a cubic equation in the domains ,  3 and  3 was presented. In this paper, without proof, we provide a solvability criterion for a cubic equation in the domains  3 \ ,  3 \  3 and  3 as a complementary part of our previous research.…”