狄利克雷单位定理的历史演变
The Historical Evolution of Dirichlet's Unit Theorem
孟祥蕊 1王淑红1
作者信息
- 1. 河北师范大学数学科学学院,石家庄 050024
- 折叠
摘要
狄利克雷单位定理是代数数论中基本的有限性定理之一,描述了任意代数数域中单位群的基本结构,在代数数论史上占有举足轻重的地位.本文通过文献考证和概念分析,对该定理的起源、演变过程进行了研究,得出如下认识:由于对算术级数中素数定理的研究,狄利克雷逐步给出了单位定理及其一般性证明,后经其他数学家的推广,最终呈现出狄利克雷单位定理的现代表述形式,即任意代数数域的单位群等于其单位根群和一个秩有限的自由阿贝尔群的直积.
Abstract
Dirichlet's unit theorem is one of the fundamental finiteness theorems in algebraic number theory.It describes the basic structure of unit groups in any algebraic number field and occupies a pivotal position in the history of algebraic number theory.Through the method of literature research and concept analysis,the origin and evolution process of the theorem are studied in this paper.The research shows that due to the research on the prime number theorem in arithmetic series,Dirichlet formulated the unit theorem and its general proof gradually.After the generalization of other mathematicians,the modern expression form of Dirichlet's unit theorem finally emerged,stating that the group of units of any algebraic number field is the direct product of its group of roots of unity and a free abelian group whose rank is finite.
关键词
狄利克雷/狄利克雷单位定理/数论/理想Key words
Johann Peter Gustav Lejeune Dirichlet/Dirichlet's unit theorem/number theory/ideal引用本文复制引用
基金项目
国家自然科学基金资助项目(12271138)
出版年
2024
NETL
NSTL
万方数据