Abstract Algebra
抽象代数是研究代数结构的现代数学分支，是纯数学的核心课程之一。课程内容包括群论（群、子群、正规子群、商群、群同态、群作用、西罗定理）、环论（环、理想、商环、环同态、多项式环）、域论（域扩张、分裂域、有限域）。学生将学习抽象的公理化方法，掌握代数结构的基本理论，培养抽象思维和逻辑推理能力。本课程在数论、代数几何、密码学、理论物理等领域有重要应用。
Abstract Algebra studies algebraic structures and is one of the core courses in pure mathematics. Topics include group theory (groups, subgroups, normal subgroups, quotient groups, homomorphisms, group actions, Sylow theorems), ring theory (rings, ideals, quotient rings, ring homomorphisms, polynomial rings), and field theory (field extensions, splitting fields, finite fields). Students will learn abstract axiomatic methods and develop abstract thinking skills.
研究西罗定理及其证明，运用西罗定理对低阶有限群进行分类。选择一个具体阶数的群（如阶数为pq或p^3），详细分析其结构，确定所有可能的同构类。探讨群作用在计数问题中的应用，研究正规子群与商群的结构。要求包含定理证明、具体例子验证和分类结果的完整呈现。
Study Sylow theorems and their proofs, apply them to classify finite groups of low order. Choose groups of a specific order (such as pq or p^3), analyze their structures in detail, and determine all possible isomorphism classes. Explore applications of group actions in counting problems.