Dive into the fundamental concepts of abstract algebra in this comprehensive 3.5-hour video course. Explore the foundations of modern algebra, starting with group theory and progressing through rings, fields, and vector spaces. Learn key definitions, theorems, and applications, including group properties, subgroups, cosets, Lagrange's theorem, homomorphisms, and isomorphisms. Examine specific group types such as cyclic, symmetric, and dihedral groups, as well as matrix groups and direct products. Delve into ring theory, covering integral domains, ideals, and field properties. Conclude with an introduction to vector spaces and modules, providing a solid foundation for further study in advanced mathematics.
Overview
Syllabus
What is Abstract Algebra? (Modern Algebra).
Group Definition (expanded) - Abstract Algebra.
Abstract Algebra: The definition of a Group.
Abstract Algebra: Motivation for the definition of a group.
Abstract Algebra: The definition of a Subgroup.
Group Multiplication Tables | Cayley Tables (Abstract Algebra).
Cosets and Lagrange’s Theorem - The Size of Subgroups (Abstract Algebra).
Normal Subgroups and Quotient Groups (aka Factor Groups) - Abstract Algebra.
Cyclic Groups (Abstract Algebra).
Abstract Algebra: Group or Not Group? (Integer edition).
Group Homomorphisms - Abstract Algebra.
Homomorphisms (Abstract Algebra).
Isomorphisms (Abstract Algebra).
The Kernel of a Group Homomorphism – Abstract Algebra.
The Order of an Element (Abstract Algebra).
Symmetric Groups (Abstract Algebra).
Cycle Notation of Permutations - Abstract Algebra.
Dihedral Group (Abstract Algebra).
Symmetry Groups of Triangles (Abstract Algebra).
Matrix Groups (Abstract Algebra).
Direct Products of Groups (Abstract Algebra).
Simple Groups - Abstract Algebra.
Abstract Algebra: The definition of a Ring.
Ring Definition (expanded) - Abstract Algebra.
Ring Examples (Abstract Algebra).
Units in a Ring (Abstract Algebra).
Integral Domains (Abstract Algebra).
Ideals in Ring Theory (Abstract Algebra).
Field Definition (expanded) - Abstract Algebra.
Abstract Algebra: The definition of a Field.
Field Examples - Infinite Fields (Abstract Algebra).
What is a Vector Space? (Abstract Algebra).
What is a Module? (Abstract Algebra).
Taught by
Socratica
