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Power Integral Bases for Q（ζ+ζ～（-1）） and Some Results in Z[-2～（1/2）],Z[2～（1/2）]

论文摘要

本文首先给出了确定分圆域的极大实子域的幂元整基的两种方法;然后利用这两种方法找出了Q(ζm+ζm-1)（m=5,7,8,9,12,16,20,24）的所有幂元整基。所得到的结果对于任意分圆域的极大实子域的幂元整基的探讨很有帮助。 本文同时证明了方程x4-y4=z2在Z[-21/2],Z[21/2]铜中只有平凡解。从而表明Z[-21/2](对应为Z[21/2])中任意一非零整数的平方都不能表示为两个非零整数的四次方的差的形式。 本文最后讨论了Z[-21/2]中任意一非零整数表示为两个整数平方差的形式的表示种数。同时也讨论了Z[-2（1/2）]中任意一非零整数x表示为z=x2+2y2（其中x,y∈Z[-2（1/2）]）的表示种数。

论文目录

  • 摘要
  • Abstract
  • Preface
  • Chapter 1 Power integral bases in maximal real subfields of some cyclotomic fields
  • § 1.1. Some Lemmas
  • § 1.2. Main Results
  • 4 - y4 = z2 in Z[（-2）1/2], Z[21/2]'>Chapter 2 The diophantine equation x4 - y4 = z2 in Z[（-2）1/2], Z[21/2]
  • 4 - y4 = z2 in Z[（-2）1/2]'>§ 2.1. The diophantine equation x4 - y4 = z2 in Z[（-2）1/2]
  • 4 - y4 = z2 in Z[21/2]'>§ 2.2. The diophantine equation x4 - y4 = z2 in Z[21/2]
  • 1/2]'>Chapter 3 The difference of two squares of integers in Z[（-2）1/2]
  • 1/2] as z = x2 - y2'>§ 3.1. Any nonzero integer z ∈ Z[（-2）1/2] as z = x2 - y2
  • 1/2] as z = x2 + 2y2'>§ 3.2. Any nonzero integer z ∈ Z[（-2）1/2] as z = x2 + 2y2
  • References

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