Abstract Algebra

Build groups from the Cayley table up: the axioms, the first proofs, the groups ℤₙ and U(n), orders of elements and subgroups.

1. Binary Operations and Cayley Tables
2. The Group Axioms
3. The Groups ℤₙ and U(n)
4. First Proofs: Uniqueness, Cancellation and Inverses
5. The Order of an Element
6. Subgroups and the Subgroup Test

Groups made by one element: generators, the orders of powers, and the tidy subgroup structure of ℤₙ.

1. Cyclic Groups and Generators
2. Orders of Powers
3. Subgroups of Cyclic Groups
4. Subgroup Lattices

Rearrangements as a group: composing permutations, cycle notation, orders and signs, and the symmetries of a regular polygon.

1. Permutations and Composition
2. Cycle Notation
3. The Order of a Permutation
4. Even and Odd Permutations
5. Dihedral Groups

Cut a group into cosets of a subgroup, prove Lagrange's theorem, and use it to limit the orders of subgroups and elements.

1. Cosets
2. Lagrange's Theorem
3. Consequences of Lagrange's Theorem

Subgroups you can divide by, maps that respect the operation, and the first isomorphism theorem that ties them together.

1. Normal Subgroups
2. Quotient Groups
3. Group Homomorphisms
4. Kernels and Images
5. Isomorphic Groups
6. Cayley's Theorem
7. The First Isomorphism Theorem

Build groups as products, decide when a product is cyclic, and list every finite abelian group of a given order.

1. Direct Products
2. Classifying Finite Abelian Groups
3. Invariant Factors and Telling Abelian Groups Apart

Let groups move things: orbits and stabilizers, conjugacy classes, Burnside's counting lemma and the Sylow theorems.

1. Group Actions
2. The Orbit–Stabilizer Theorem
3. Conjugacy Classes and the Class Equation
4. Burnside's Lemma
5. The Sylow Theorems

Two operations at once: rings, zero divisors, integral domains and fields, ideals, ring homomorphisms and quotient rings.

1. Rings
2. Integral Domains and Fields
3. Ideals
4. Ring Homomorphisms
5. Quotient Rings
6. Prime and Maximal Ideals

Polynomials with coefficients in a ring or field: division with remainder, gcds, roots and tests for irreducibility.

1. Polynomial Rings
2. Division and Roots in F[x]
3. The Euclidean Algorithm for Polynomials
4. Irreducible Polynomials of Degree 2 and 3
5. Testing Irreducibility over ℚ

Grow fields by adjoining roots: arithmetic in ℚ(√2) and F[x]/(p(x)), minimal polynomials, degrees and the finite fields GF(pⁿ).

1. Field Extensions
2. Minimal Polynomials and Degree
3. Degrees and the Tower Law
4. Finite Fields GF(pⁿ)
5. The Multiplicative Group of a Finite Field