A Concrete Introduction to Higher Algebra

This book is an informal and readable introduction to higher algebra at thepostcalculus level. The concepts of ring and field are introduced throughstudy of the... » 

This book is an informal and readable introduction to higher algebra at thepostcalculus level. The concepts of ring and field are introduced throughstudy of the familiar examples of the integers and polynomials. A strongemphasis on congruence classes leads in a natural way to finite groups andfinite fields. The new examples and theory are built in a wellmotivatedfashion and made relevant by many applications to cryptography codingintegration history of mathematics and especially to elementary andcomputational number theory. The later chapters include expositions of Rabiinsprobabilistic primality test quadratic reciprocity and the classification offinite fields. Over 900 exercises ranging from routine examples to extensionsof theory are found throughout the book hints and answers for many of themare included in an appendix. At every stage a wide variety of applications ispresented...The userfriendly exposition is appropriate for the intendedaudience. T.W. Hungerford The new edition will contain topics such as Luhnsformula homomorphisms detaching coefficients and more. TOCNumbers.Induction. Euclids Algorithm. Unique Factorization. Congruences.Congruence Classes. Applications of Congruences. Rings and Fields. Fermatsand Eulers Theorems. Applications of Fermats and Eulers Theorems. OnGroups. The Chinese Remainder Theorem. Matrices and Codes. Polynomials.Unique Factorization. The Fundamental Theorem of Algebra. Derivatives.Factoring in QxI. The Binomial Theorem in Characteristic p. Congruencesand the Chinese Remainder Theorem. Applications of the Chinese RemainderTheorem. Factoring in Fpx and in Zx. Primitive Roots. Cyclic Groups andPrimitive Roots. Pseudoprimes. Roots of Unity in ZmZ. Quadratic Residues.Congruence Classes Modulo a Polynomial. Some Applications of Finite Fields.Classifying Finite Fields. «