分圆域Q(ζ_(15))的幂元整基(英文)  

Power Bases for the Cyclotomic Field Q(ζ_(15))

作　　者：夏建国[1] 汪少祖[1] 

机构地区：[1]南京师范大学数学与计算机科学学院,江苏南京210097

出　　处：《四川师范大学学报（自然科学版）》2005年第6期664-666,共3页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(10471118)资助项目~~

摘　　要：称一个伽罗华数域L有一个幂元整基,如果它的代数整数环具有形式Z[α],其中α∈L.并且此时称α为幂元整基的生成元.两个幂元整基的生成元α和α′称为等价的,如果α′=m±σ(α),其中m∈Z并且σ∈Gal(L/Q).讨论了分圆域Q(ζ15)的幂元整基的生成元,其中ζ15是15次本原单位根.众所周知ζ15,(1-ζ15)-1和(1+ζ15)-1都是分圆域Q(ζ15)的幂元整基的生成元.证明了当α+α-Z时α是分圆域Q(ζ)的幂元整基的生成元当且仅当α与ζ等价.A Galois number field L is said to have a power basis if its ring of integers is of the form Z[α] for some α∈L. In this case a is called a generator of power basis in L. Let a and α＇ be generators of two power bases in L, a and a＇ is called equivalent if α＇=m±σ（α） for some m∈Z and σ∈Gal（L/Q）. In this paper, we discuss the generators of power bases in the cyclotomic field Q（ξ15）, where ξ15 is a primitive 15-th root of unity. It is well-know that ξ15, （1-ξ15）^-1 and （1＋ξ15）^-1 are all generators of power bases in the cyclotomic field Q（ξ15）. We show that if α＋α^- Z then a is a generator of power basis in the cyclotomic field Q（ξ15） if and only if a is equivalent to ξ15.

关 键 词：分圆域 生成元 幂元整基 

分 类 号：O156.2[理学—数学]