Course Overview

Advanced study of abstract algebraic structures. Key areas included:

Group Theory

• Group axioms and properties
• Subgroups and normal subgroups
• Cyclic and abelian groups
• Group actions and orbits

Ring Theory

• Ring axioms and properties
• Ideals and quotient rings
• Polynomial rings
• Principal ideal domains

Field Theory

• Field extensions
• Finite fields
• Galois theory introduction
• Applications in geometry

Homomorphisms and Isomorphisms

• Structure-preserving maps
• Isomorphism theorems
• Direct products
• Factor groups

Applications

• Cryptography
• Coding theory
• Symmetry groups
• Algebraic structures in physics