Abstract Algebra I

Prerequisites:

None

Course description:

This course is designed for first-year students planning to major in fundamental or applied mathematics. It also welcomes students with an interest in modern mathematics. The objective is to provide a firm foundation in the core concepts and language of abstract algebra, preparing students for advanced study. The curriculum explores essential algebraic structures—such as groups, rings, fields, and modules. Instructions will blend lectures, class discussions, problem-solving sessions, and guided independent reading.

Learning objectives:

1. Understand Core Structures. Achieve a firm understanding of the defining rules, key properties, and major examples of groups, rings, and fields, mastering the fundamental language of abstract algebra.

2. Develop Proof and Analysis Skills. Build the skill to write and understand formal proofs. Gain proficiency with essential tools, including homomorphisms, quotient structures, and isomorphism theorems to analyze and relate algebraic systems.

3. Synthesize Concepts. Integrate knowledge to see the connections and differences between algebraic structures. Learn to connect abstract theory with concrete examples, creating a strong foundation for advanced mathematical study.

Detailed topics covered:

1. Group Theory. We begin with the algebraic study of symmetry, covering the axioms of a group, fundamental examples (cyclic, symmetric, and dihedral groups), and key concepts like subgroups, homomorphisms, and normal subgroups. A central achievement is understanding quotient groups and the First Isomorphism Theorem, which connects homomorphisms to these quotient structures.

2. Ring and Field Theory. Generalizing the arithmetic of integers and polynomials, we examine rings, integral domains, and fields. Core topics include the study of polynomial rings R[x] as a primary example, the definitions of prime and maximal ideals, and ring homomorphisms. We investigate important classes of rings such as Unique Factorization Domains (UFDs) and Principal Ideal Domains (PIDs), highlighting the parallels between factoring integers and polynomials. The Chinese Remainder Theorem is studied as a key result about systems of congruences and the structure of quotient rings. The Isomorphism Theorems for rings are developed, solidifying the analogy with group theory.

3. Module Theory. The course culminates with modules, which generalize vector spaces by allowing rings instead of fields as scalars. We cover the definitions of modules and submodules, homomorphisms, and quotient modules. The sequence concludes with the powerful Structure Theorem for Finitely Generated Modules over a Principal Ideal Domain (PID), which classifies such modules and demonstrates a profound synthesis of group and ring theory.