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.

The companion title, Linear Algebra, has sold over 8,000 copies

The writing style is very accessible

The material can be covered easily in a one-year or one-term course

Includes Noah Snyder's proof of the Mason-Stothers polynomial abc theorem

New material included on product structure for matrices including descriptions of the conjugation representation of the diagonal group