Undergraduate Algebra

I The Integers.- 1. Terminology of Sets.- 2. Basic Properties.- 3. Greatest Common Divisor.- 4. Unique Factorization.- 5. Equivalence Relations and Congruences.- II Groups.- 1. Groups and Examples.- 2. Mappings.- 3. Homomorphisms.- 4. Cosets and Normal Subgroups.- 5. Permutation Groups.- 6. Cyclic Groups.- 7. Finite Abelian Groups,.- III Rings.- 1. Rings.- 2. Ideals.- 3. Homomorphisms.- 4. Quotient Fields.- IV Polynomials.- 1. Euclidean Algorithm.- 2. Greatest Common Divisor.- 3. Unique Factorization.- 4. Partial Fractions.- 5. Polynomials over the Integers.- 6. Transcendental Elements.- 7. Principal Rings and Factorial Rings.- V Vector Spaces and Modules.- 1. Vector Spaces and Bases.- 2. Dimension of a Vector Space.- 3. Matrices and Linear Maps.- 4. Modules.- 5. Factor Modules.- 6. Free Abelian Group.- VI Some Linear Groups.- 1. The General Linear Group.- 2. Structure of GL2(F).- 3. SL2(F).- VII Field Theory.- 1. Algebraic Extensions.- 2. Embeddings'.- 3. Splitting Fields'.- 4. Galois Theory.- 5. Quadratic and Cubic Extensions.- 6. Solvability by Radicals.- 7. Infinite Extensions.- VIII Finite Fields.- 1. General Structure.- 2. The Frobenius Automorphism.- 3. The Primitive Elements.- 4. Splitting Field and Algebraic Closure.- 5. Irreducibility of the Cyclotomic Equation over Q.- 6. Where Does It All Go? Or Rather, Where Does Some of It Go?.- IX The Real and Complex Numbers.- 1. Ordering of Rings.- 2. Preliminaries.- 3. Construction of the Real Numbers.- 4. Decimal Expansions.- 5. The Complex Numbers.- X Sets.- 1. More Terminology.- 2. Zona's Lemma.- 3. Cardinal Numbers.- 4. Well-ordering.- 1. The Natural Numbers.- 2. The Integers.- 3. Infinite Sets.