Several Diophantine Equations in Some Rings of Integers of Quadratic Imaginary Fields

• 来源期刊: 代数集刊（英文版）
• 关键词: Diophantine equation rings of algebraic integers K2-group
• 年，卷（期）: 2007,（4）
• 页码范围: 661-668
• 页数: 8页
• 语种: 英文

摘要

In this paper, it is proved that the Diophantine equation x4 - y4 = z2 has no non-trivial coprime solutions in the rings of integers of quadratic imaginary fields Q(√-d)for d=11, 19, 43, 67, 163, which implies that the Fermat equation x4+y4=z4 has no nontrivial solutions in these fields either. Then all the solutions of the Pocklington equation x4-x2y2+y4 = (-1)σz2(σ=0 or 1) in the ring of integers of Q(√-11) are determined,and as an application, the result is applied to K2 of a field.