代數數論二

In this semester, we will introduce Class Field Theory and present its proof. Course keywords: Brauer groups, central simple algebras, local class field theory, global class field theory, L-functions 課程說明(Course Description): We will introduce Class Field Theory and give a proof based on the method of Brauer groups of local and global fields. The prerequisites are graduate algebra, harmonic analysis on LCAs, and basic algebraic number theory.

□ 具備發現問題、分析問題並解決問題的能力 Having ability to discover, analyze and solve problems. □ 培養運用數學作為發展其他學科的能力 Developing ability to interact mathematics with other subjects. □ 培養數學邏輯思考的能力 Developing ability of using mathematical logic to think. □ 學習進階數學的能力 Ability to learn advanced mathematical knowledge.

教學內容(Syllabus): 1. Introduction (i) Statements of class field theory (ii) Fundamental consequences and examples 2. Brauer groups (i) Central simple algebras (ii) Brauer groups of local fields (iii) Brauer groups of global fields 3. Proof of class field theory (i) Local class field theory (ii) Zeta functions of simple algebras (iii) Global class field theory 4. Related topics (i) Chebotarev density theorem (ii) Artin L-functions

評分方式(Evaluation): 報告 100% AI使用規則(AI Usage Policy): 開放使用AI,但嚴禁抄襲AI生成的資料。

參考書籍(References): Artin-Tate, Class Field Theory Cassels-Frohlich, Algebraic Number Theory Kato-Kurokawa-Saito, Number Theory 2, Introduction to Class Field Theory Lang, Algebraic Number Theory Milne, Class Field Theory Neukirch, Algebraic Number Theory Weil, Basic Number Theory