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An informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials, with much emphasis placed on congruence classes leading the way to finite groups and finite fields. New examples and theory are integrated in a well-motivated fashion and made relevant by many applications -- to cryptography, coding, integration, history of mathematics, and especially to elementary and computational number theory. The later chapters include expositions of Rabiin's probabilistic primality test, quadratic reciprocity, and the classification of finite fields. Over 900 exercises, ranging from routine examples to extensions of theory, are scattered throughout the book, with hints and answers for many of them included in an appendix.

A Concrete Introduction to Higher Algebra

1 Numbers 2 Induction 3 Euclid's Algorithm 4 Unique Factorization 5 Congruences 6 Congruence Classes 7 Applications of Congruences 8 Rings and Fields 9 Fermat's and Euler's Theorems 10 Applications of Fermat's and Euler's Theorems 11 On Groups 12 The Chinese Remainder Theorem 13 Matrices and Codes 14 Polynomials 15 Unique Factorization 16 The Fundamental Theorem of Algebra 17 Derivatives 18 Factoring in Q[x],I 19 The Binomial Theorem in Characteristic p 20 Congruences and the Chinese Remainder Theorem 21 Applications of the Chinese Remainder Theorem 22 Factoring in Fp[x] and in Z[x] 23 Primitive Roots 24 Cyclic Groups and Primitive Roots 25 Pseudoprimes 26 Roots of Unity in Z/mZ 27 Quadratic Residues 28 Congruence Classes Mopdulo a Polynomial 29 Some Applications of Finite Fields 30 Classifying Finite Fields