An Informal Introduction to Abstract Algebra

Explore an informal introduction to abstract algebra in this 49-minute lecture, focusing on concepts relevant to algebraic topology. Gain insights into fields, rings, and vector spaces through descriptive explanations and key examples. Begin with fundamental number systems like natural numbers, integers, and rational numbers. Examine fields, using rational numbers as the primary example, and delve into complex rational numbers, finite fields, and algebraic extensions. Investigate rings through examples such as integers, polynomials, and square matrices. Study vector spaces, including row vectors, polynomials of limited degree, and matrices with specific operations. Conclude by exploring three essential algebraic constructions: subobjects, homomorphisms, and quotient objects. Prepare for a subsequent lecture on commutative and non-commutative groups in algebraic topology.

Introduction
General patterns
Natural numbers
Integers
Other examples
Noncommutative ring
Fields
Finite fields
Field properties
Vector spaces
Other vector spaces
Common themes