Unit Code: MTH802

Unit Name: Advanced Abstract Algebra

Description: Algebra is the language of modern mathematics. This course introduces students to that language through a study of Group Basic Axioms and examples, Dehedral Groups, Symmetric Groups, Matrix Groups and The Quaternion Group, Quotien Groups and Homomorphisms, Automorphisms, The Sylow Theorems, Direct Products, The Fundamental Theorem of Finitely Generated Abelian Groups, Table of Groups of Small Order, Semidirect products, p- groups, Nilpotent Groups, and solvable Groups, Application in Groups of Medium order, Ring Basic Definitions and Examples, Ring Homomorphisms and Quotient Rings, Properties of Ideals, Rings of Fractions, Euclidean Domains, principal Ideal Domains and Unique Factorization Domains, Polynomial Rings, Basic Theory of field Extensions, Algebraic Extensions, The Fundamental Theorem of Galois Theory and Finite Fields. Composite extensions and Simple Extensions. Cyclotomic Extensions and Abelian Extensions over Q. Galois Groups of Polynomials. Solvable and Radical Extensions: Insolvability of the Quintic. Computation of Galois Groups over Q. Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups.

Learning Target Outcomes: As a result of successfully completing this course, the student will be able to: 1. Utilize advanced abstract algebra such as groups, rings and fields and their role in modern mathematics and applied contexts; 2. Demonstrate accurate and efficient use of advanced algebraic techniques; 3. Provide different algebraic structures and examples of specific constructs; 4. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from advanced algebra; 5. Apply problem-solving using advanced algebraic techniques applied to diverse situations in computer science and other mathematical contexts.

Prerequisite: Minimum Entry Requirements of the programme

Prerequisite Sentence: Minimum Entry Requirement of the programme

Credit Point: 30

Offered In: Semester 1,2