I INTEGERS.- 1 Numbers.- 2 Induction; the Binomial Theorem.- A. Induction.- B. Another Form of Induction.- C. Well-ordering.- D. The Binomial Theorem.- 3 Unique Factorization into Products of Primes.- A. Euclid's Algorithm.- B. Greatest Common Divisors.- C. Unique Factorization.- D. Exponential Notation; Least Common Multiples.- 4 Primes.- A. Euclid.- B. Some Analytic Results.- C. The Prime Number Theorem.- 5 Bases.- A. Numbers in Base a.- B. Operations in Base a.- C. Multiple Precision Long Division.- D. Decimal Expansions.- 6 Congruences.- A. Definition of Congruence.- B. Basic Properties.- C. Divisibility Tricks.- D. More Properties of Congruence.- E. Congruence Problems.- F. Round Robin Tournaments.- 7 Congruence Classes.- 8 Rings and Fields.- A. Axioms.- B. ?m.- 9 Matrices and Vectors.- A. Matrix Multiplication.- B. The Ring of n × n Matrices.- C. Linear Equations.- D. Determinants and Inverses.- E. Row Operations.- F. Subspaces, Bases, Dimension.- 10 Secret Codes, I.- 11 Fernjat's Theorem, I: Abelian Groups.- A. Fermat's Theorem.- B. Abelian Groups.- C. Euler'sr Theorem.- D. Finding High Powers mod m.- E. The Order of an Element.- F. About Finite Fields.- G. Nonabelian Groups.- 12 Repeating Decimals, I.- 13 Error Correcting Codes, I.- 14 The Chinese Remainder Theorem.- A. The Theorem.- B. A Generalization of Fermat's Theorem.- 15 Secret Codes, II.- II POLYNOMIALS.- 1 Polynomials.- 2 Unique Factorization.- A. Division Theorem.- B. Greatest Common Divisors.- C. Factorization into Irreducible Polynomials.- 3 The Fundamental Theorem of Algebra.- A. Irreducible Polynomials in ?[x].- B. Proof of the Fundamental Theorem.- 4 Irreducible Polynomials in ?[x].- 5 Partial Fractions.- A. Rational Functions.- B. Partial Fractions.- C. Integrating.- D. A Partitioning Formula.- 6 The Derivative of a Polynomial.- 7 Sturm's Algorithm.- 8 Factoring in ?[x], I.- A. Gauss's Lemma.- B. Finding Roots.- C. Testing for Irreducibility.- 9 Congruences Modulo a Polynomial.- 10 Fermat's Theorem, II.- A. The Characteristic of a Field.- B. Applications of the Binomial Theorem.- 11 Factoring in ?;[x], II: Lagrange Interpolation.- A. The Chinese Remainder Theorem.- B. The Method of Lagrange Interpolation.- 12 Factoring in ?p[x].- 13 Factoring in ?[x], III: Mod m.- A. Bounding the Coefficients of Factors of a Polynomial.- B. Factoring Modulo High Powers of Primes.- III FIELDS.- 1 Primitive Elements.- 2 Repeating Decimals, II.- 3 Testing for Primeness.- 4 Fourth Roots of One in ?p.- A. Primes.- B. Finite Fields of Complex Numbers.- 5 Telephone Cable Splicing.- 6 Factoring in ?[x], IV: Bad Examples Modp.- 7 Congruence Classes Modulo a Polynomial: Simple Field Extensions.- 8 Polynomials and Roots.- A. Inventing Roots of Polynomials.- B. Finding Polynomials with Given Roots.- 9 Error Correcting Codes, II.- 10 Isomorphisms, I.- A. Definitions.- B. Examples Involving ?.- C. Examples Involving F[x].- D. Automorphisms.- 11 Finite Fields are Simple.- 12 Latin Squares.- 13 Irreducible Polynomials in ?p[x].- A. Factoring xpn ? x.- B. Counting Irreducible Polynomials.- 14 Finite Fields.- 15 The Discriminant and Stickelberger's Theorem.- A. The Discriminant.- B. Roots of Irreducible Polynomials in ?c[x].- C. Stickelberger's Theorem.- 16 Quadratic Residues.- A. Reduction to the Odd Prime Case.- B. The Legendre Symbol.- C. Proof of the Law of Quadratic Reciprocity.- 17 Duplicate Bridge Tournaments.- A. Hadamard Matrices.- B. Duplicate Bridge Tournaments.- C. Bridge for 8.- D. Bridge forp + 1.- 18 Algebraic Number Fields.- 19 Isomorphisms, II.- 20 Sums of Two Squares.- 21 On Unique Factorization.- Exercises Used in Subsequent Chapters.- Comments on the Starred Problems.- References.

This book is written as an introduction to higher algebra for students with a background of a year of calculus. The book developed out of a set of notes for a sophomore-junior level course at the State University of New York at Albany entitled Classical Algebra. In the 1950s and before, it was customary for the first course in algebra to be a course in the theory of equations, consisting of a study of polynomials over the complex, real, and rational numbers, and, to a lesser extent, linear algebra from the point of view of systems of equations. Abstract algebra, that is, the study of groups, rings, and fields, usually followed such a course. In recent years the theory of equations course has disappeared. Without it, students entering abstract algebra courses tend to lack the experience in the algebraic theory of the basic classical examples of the integers and polynomials necessary for understanding, and more importantly, for ap­ preciating the formalism. To meet this problem, several texts have recently appeared introducing algebra through number theory.