A Concrete Introduction to Higher Algebra

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Springer Science & Business Media, 26 нояб. 2008 г. - Всего страниц: 604
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This book is 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. The new examples and theory are built 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 are found throughout the book.

 

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Содержание

Numbers
3
Induction
8
Euclids Algorithm
27
Unique Factorization
53
Congruence
71
Congruence classes and rings
90
Congruence Classes
93
Rings and Fields
122
Congruences and the Chinese Remainder Theorem
355
Fast Polynomial Multiplication
372
Primitive Roots
384
Cyclic Groups and Cryptography
385
Carmichael Numbers
413
Quadratic Reciprocity
433
Quadratic Applications
458
Finite Fields
476

Matrices and Codes
147
Congruences and Groups
168
Fermats and Eulers Theorems
169
Applications of Eulers Theorem
201
Groups
223
The Chinese Remainder Theorem
253
Polynomials
282
Polynomials
285
Unique Factorization
294
The Fundamental Theorem of Algebra
307
Polynomials in Qx
339
Congruence Classes Modulo a Polynomial
477
Homomorphisms and Finite Fields
495
BCH Codes
511
Factoring Polynomials
528
Factoring in Zx
531
Irreducible Polynomials
557
Answers and Hints to the Exercises
568
References
595
Index
599
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