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Abstract Algebra

Groups, rings, fields: symmetry made into structure.

Lessons

Introductory Groups The axioms, examples (integers mod n, symmetries, permutations), order of an element. 3^100 mod 7 Core Subgroups, cosets and Lagrange's theorem Why the order of a subgroup divides the order of the group. divisors of 12 Core Cyclic groups and permutation groups Generators, orders, cycles, transpositions, parity. inverse of 5 mod 12 Core Homomorphisms, normal subgroups and quotient groups Structure-preserving maps, kernels, and the first isomorphism theorem. 12 mod 5 Core Rings and fields Two operations: integers, polynomials, ℤ/n, and when division is possible. inverse of 7 mod 11 Advanced Galois theory: why the quintic has no formula Field extensions, the Galois group, and solvability by radicals. x^3 + x + 1 = 0

Take the rules that integers obey, keep only the rules, and ask what else obeys them. Groups capture symmetry, rings capture arithmetic, fields capture solvability. Galois used them to prove the quintic has no formula.

More lessons

Preliminaries

A Short Note on Proofs Sets and Equivalence Relations Sage

The Integers

Mathematical Induction The Division Algorithm Sage

Cyclic Groups

Cyclic Subgroups Multiplicative Group of Complex Numbers The Method of Repeated Squares Sage

Permutation Groups

Definitions and Notation Dihedral Groups Sage

Cosets and Lagrange's Theorem

Cosets Lagrange's Theorem Fermat's and Euler's Theorems Sage

Introduction to Cryptography

Private Key Cryptography Public Key Cryptography Sage

Algebraic Coding Theory

Error-Detecting and Correcting Codes Linear Codes Parity-Check and Generator Matrices Efficient Decoding Sage

Isomorphisms

Definition and Examples Direct Products Sage

Normal Subgroups and Factor Groups

Factor Groups and Normal Subgroups The Simplicity of the Alternating Group Sage

Homomorphisms

Group Homomorphisms The Isomorphism Theorems Sage

Matrix Groups and Symmetry

Matrix Groups Symmetry

The Structure of Groups

Finite Abelian Groups Solvable Groups Sage

Group Actions

Groups Acting on Sets The Class Equation Burnside's Counting Theorem Sage

The Sylow Theorems

The Sylow Theorems Examples and Applications Sage

Rings

Rings Integral Domains and Fields Ring Homomorphisms and Ideals Maximal and Prime Ideals An Application to Software Design Sage

Polynomials

Polynomial Rings The Division Algorithm Irreducible Polynomials Sage

Integral Domains

Fields of Fractions Factorization in Integral Domains Sage

Lattices and Boolean Algebras

Lattices Boolean Algebras The Algebra of Electrical Circuits Sage

Vector Spaces

Definitions and Examples Subspaces Linear Independence Sage

Fields

Extension Fields Splitting Fields Geometric Constructions Sage

Finite Fields

Structure of a Finite Field Polynomial Codes Sage

Galois Theory

Field Automorphisms The Fundamental Theorem Applications Sage

Rings and fields

Ring (mathematics)Ideal (ring theory)Quotient ring Integral domain Polynomial ring Unique factorization domain Principal ideal domain Field (mathematics)Field extension Finite field Splitting field Algebraically closed field Characteristic (algebra)

Galois and beyond

Galois group Solvable group Abel–Ruffini theorem Module (mathematics)Representation theory Lie algebra Category of groups Universal algebra

Some of these lessons are adapted from open texts: Judson, Abstract Algebra: Theory and Applications, Wikipedia. Each lesson page names its source and licence.

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

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