Abstract Algebra Ii: The Next Step In Algebraic Thinking

aster the deeper layers of algebraic theory with a focus on rings, fields, and polynomial symmetries.

What you'll learn

How groups act on sets, and how orbits, stabilizers, and isotropy subgroups reveal group structure

The Class Equation, Cauchy’s Theorem, and the Sylow Theorems—cornerstones of finite group theory

The concept of automorphisms, including inner automorphisms and automorphisms of abelian groups

How to classify and analyze simple groups, including nonabelian examples and group orders

The structure and behavior of rings, including subrings, matrix rings, polynomial rings, and group rings

Deep understanding of integral domains, principal ideal domains (PIDs), Euclidean domains, and unique factorization domains (UFDs)

The theory of ideals, including prime, maximal, and principal ideals, and the isomorphism theorems for rings

How to construct and work with fields, division rings, and field extensions

The logic behind algebraic closures, constructible numbers, and splitting fields

The structure of finite fields, Galois fields, and their applications in coding theory

The fundamentals of Galois theory, including automorphisms, fixed fields, and the Fundamental Theorem of Galois Theory

How to determine solvability by radicals, and the connection between group theory and polynomial equations

An introduction to lattices, Boolean algebras, and their algebraic properties and applications

Requirements

Completion of Abstract Algebra I or equivalent knowledge of groups, rings, and basic proofs

Familiarity with set theory, functions, and basic linear algebra

A willingness to engage with abstract reasoning and formal mathematical logic

Description

Abstract Algebra II: Group Actions, Rings, Fields & Galois TheoryExplore the Deep Structures of Algebra That Shape Modern MathematicsAbstract Algebra II is not just a continuation—it's a transformation in how you understand mathematics. If Abstract Algebra I introduced you to the foundational concepts of groups, rings, and fields, this course takes you into the core of algebraic reasoning, where structure, symmetry, and abstraction converge.This is the mathematics that underpins cryptography, coding theory, quantum mechanics, and algebraic geometry. It’s the language of automorphisms, field extensions, and Galois groups—tools that mathematicians use to solve equations that defy classical methods and to understand the deep relationships between algebraic objects.You’ll begin with group actions, a powerful framework for understanding how groups interact with sets, leading to insights about symmetry, orbits, and stabilizers. From there, you’ll explore automorphisms, the internal symmetries of algebraic structures, and how they relate to the Class Equation, Sylow Theorems, and the classification of simple groups.Then, the course shifts into ring theory, where you’ll study subrings, ideals, and homomorphisms, and discover how structures like principal ideal domains (PIDs) and Euclidean domains govern factorization and divisibility. You’ll learn how polynomial rings behave over different domains, and how tools like Gauss’ Lemma and Eisenstein’s Criterion help identify irreducible elements.The second half of the course is devoted to field theory and Galois theory—the crown jewel of classical algebra. You’ll explore field extensions, splitting fields, and finite fields, and learn how Galois groups encode the solvability of polynomials. You’ll see how solvability by radicals connects group theory to the age-old question of solving equations, and how constructible numbers relate to geometric problems like trisecting angles and squaring the circle.Finally, the course introduces lattices and Boolean algebras, bridging algebra with logic and computer science. These structures reveal how algebraic reasoning applies to circuits, decision-making, and symbolic computation.Why Take This Course?To build on your foundation from Abstract Algebra I and master advanced algebraic structuresTo prepare for graduate-level mathematics, research, or competitive examsTo understand the algebra behind modern applications in cryptography, coding theory, and theoretical physicsTo develop mathematical maturity through rigorous proofs, abstract reasoning, and structural thinkingTo connect algebra with geometry, logic, and computation in a unified framework

Overview

Section 1: Group Actions

Lecture 1 Group Actions

Lecture 2 Examples of Group Actions

Lecture 3 Orbits of a Group Action

Section 2: Isotropy Subgroups

Lecture 4 More Examples of Group Actions

Lecture 5 Strong Cayley's Theorem

Lecture 6 Stable Sets and Isotropy Subgroups

Lecture 7 The Fundamental Counting Principle (Group Actions)

Section 3: Automorphisms

Lecture 8 Automorphisms (Group Theory)

Lecture 9 Automorphisms of Abelian Groups

Lecture 10 Inner Automorphisms

Section 4: The Class Equation

Lecture 11 The Class Equation

Lecture 12 p-Groups

Section 5: Burnside's Theorem

Lecture 13 Burnside's Theorem

Lecture 14 Counting Orbits Using Burnside's Theorem

Section 6: Cauchy's Theorem

Lecture 15 Cauchy's Theorem

Lecture 16 The First Sylow Theorem

Section 7: The Sylow Theorems

Lecture 17 Lemmas For Sylow Theory

Lecture 18 The Second Sylow Theorem

Lecture 19 The Third Sylow Theorem

Section 8: Simple Groups

Lecture 20 Simple Groups

Section 9: Sylow Theory and Simple Groups

Lecture 21 Groups of Order pq

Lecture 22 The Hunt for Nonabelian Simple Groups: Part 1 – Groups of Order pⁿ and pq

Lecture 23 The Hunt for Nonabelian Simple Groups: Part 2 – Groups of Order (pⁿ)k: 18, 20, 2

Lecture 24 The Hunt for Nonabelian Simple Groups: Part 3 – Groups of Order (2ⁿ)p: 12, 56

Lecture 25 The Hunt for Nonabelian Simple Groups: Part 4 – Groups of Order pqr: 30, 42

Lecture 26 The Hunt for Nonabelian Simple Groups: Part 5 – Groups of Order 12k: 24, 36, 48

Section 10: Rings

Lecture 27 Rings (Abstract Algebra)

Lecture 28 Subrings

Lecture 29 The Dominance of Zero in a Ring

Lecture 30 Matrix Rings

Lecture 31 Polynomial Rings

Lecture 32 Group Rings

Lecture 33 Fields (Abstract Algebra)

Lecture 34 Division Rings

Lecture 35 Units (Ring Theory)

Section 11: Integral Domains

Lecture 36 Integral Domains

Lecture 37 Gaussian Integers

Lecture 38 Cancellation in Integral Domains

Lecture 39 The Characteristic of a Ring

Section 12: Ring Homomorphisms

Lecture 40 Ring Homomorphisms

Lecture 41 Examples of Ring Homomorphisms

Lecture 42 Kernels of Ring Homomorphisms

Section 13: Ideals

Lecture 43 The Isomorphism Theorems (Ring Theory)

Lecture 44 Principal Ideals

Lecture 45 Maximal Ideals

Lecture 46 Prime Ideals

Section 14: Field of Fractions

Lecture 47 The Field of Fractions

Section 15: Unique Factorization Domains

Lecture 48 Factorization and Divisibility (Ring Theory)

Lecture 49 Unique Factorization Domains

Lecture 50 An Irreducible Element that is Not a Prime

Lecture 51 Irreducibles are Prime in Integral Domains

Lecture 52 Common Divisors and Multiples (Ring Theory)

Section 16: Principal Ideal Domains

Lecture 53 Principal Ideal Domains

Lecture 54 Noetherian Domains

Lecture 55 PID's are UFD's

Section 17: Euclidean Domains

Lecture 56 Norms of an Integral Domains

Lecture 57 Euclidean Domains

Lecture 58 Euclidean Domains are PID's

Lecture 59 Multiplicative Norms in a Euclidean Domain

Section 18: Polynomial Rings

Lecture 60 Polynomial Rings (Reprise)

Lecture 61 The Degree Function of a Polynomial Ring

Lecture 62 Multivariate Polynomial Rings

Lecture 63 Polynomial Rings are Euclidean Domains

Section 19: Gauss' Lemma

Lecture 64 Polynomial Division over a Field

Lecture 65 The Factor Theorem over a Field

Lecture 66 Gauss' Lemma

Section 20: Irreducible Polynomials

Lecture 67 The Polynomial Ring over a UFD is a UFD

Lecture 68 Irreducible Polynomials of Small Degree

Lecture 69 Eisenstein's Criterion

Section 21: Linear Algebra and Modules

Lecture 70 R-Modules

Lecture 71 Submodules

Section 22: Zorn's Lemma

Lecture 72 Zorn's Lemma

Lecture 73 The Expansion and Pruning Theorems (Linear Algebra)

Lecture 74 The Basis Theorem

Section 23: Field Extensions

Lecture 75 Field Extensions

Lecture 76 Kronecker’s Theorem

Lecture 77 The Field of Order 4

Section 24: Algebraic Extensions

Lecture 78 Examples of Radical Extensions

Lecture 79 Finite Extensions

Lecture 80 Degrees of a Field Extension

Section 25: Algebraic Closures

Lecture 81 The Subset of Algebraic Elements is a Subfield

Lecture 82 Algebraically Closed Fields

Lecture 83 Algebraic Closures

Section 26: Constructible Numbers

Lecture 84 Constructible Numbers

Lecture 85 Doubling the Cube

Lecture 86 Squaring the Circle

Lecture 87 Trisecting the Angle

Lecture 88 Constructible Regular Polygons

Section 27: Splitting Fields

Lecture 89 Splitting Fields

Lecture 90 Finite Fields are Splitting Fields

Section 28: Finite Fields

Lecture 91 Galois Fields

Lecture 92 Primitive Roots of Finite Fields

Lecture 93 Review of Linear Codes

Section 29: Polynomial Codes

Lecture 94 Cyclic Codes

Lecture 95 Polynomial Codes

Lecture 96 Polynomial Codes and Group Rings

Lecture 97 Minimal Generator Polynomial

Section 30: BCH Codes

Lecture 98 Roots of Unity and Finite Fields

Lecture 99 Roots of Unity and Cyclic Codes

Lecture 100 BCH Codes

Section 31: Field Automorphisms

Lecture 101 Automorphisms and Category Theory

Lecture 102 Separable and Galois Extensions of Fields

Lecture 103 Conjugates (Field Theory)

Lecture 104 Order of Galois Groups

Lecture 105 Galois Groups of Finite Fields

Section 32: The Fundamental Theorem of Galois

Lecture 106 Fixed Fields

Lecture 107 The Fundamental Theorem of Galois

Lecture 108 Biquadratic Extensions

Lecture 109 Cyclotomic Extensions

Section 33: Galois Groups of Polynomials

Lecture 110 Galois Groups of Polynomials

Lecture 111 Galois Group of Quartic Polynomial which is Dihedral

Lecture 112 Galois Group of Quintic Polynomial which is Symmetric

Section 34: Solvability by Radicals

Lecture 113 Solvable Groups

Lecture 114 Solvability by Radicals

Section 35: Lattices

Lecture 115 Semilattices

Lecture 116 Lattices

Section 36: Boolean Algebras

Lecture 117 Bounded Lattices

Lecture 118 Distributive Lattices

Lecture 119 Boolean Algebras

Lecture 120 Properties of Boolean Algebras

Lecture 121 De Morgan's Laws (Boolean Algebra)

Lecture 122 Finite Boolean Algebras

Lecture 123 Boolean Algebras and Electric Circuits

Students who completed Abstract Algebra I and want to continue their studies,Learners preparing for graduate-level mathematics, math competitions, or research,Computer scientists, physicists, and engineers interested in algebraic structures and coding theory,Anyone passionate about mathematical abstraction and structure