Preface * Synopsis * PART 1: A Fundamental Theorem * General Arithmetic * A Fundamental Theorem * Roots Field (Simple Algebraic Extensions) * Factorization of Polynomials with Integer Coefficients * A Factorization Algorithm * Validation of the Factorization Algorithm * About the Factorization Algorithm * Proof of the Fundamental Theorem * Minimal Splitting Polynomials * PART 2: Topics in Algebra * Galois' Fundamental Theorem * Algebraic Quantities * Adjunctions and the Factorization of Polynomials * Symmetric Polynomials and the Splitting Field of x^n + c_1x^{n-1} + ... + c_n * A Fundamental Theorem of Divisor Theory * PART 3: Some Quadratic Problems * Hypernumbers * Modules * The Class Semigroup * Multiplication of Modules and Module Classes * Is A a Square Mod p? * Gauss's Composition of Forms * The Construction of Compositions * PART 4: The Genus of an Algebraic Curve * Abel's Memoir * Euler's Addition Formula * An Algebraic Definition of the Genus * Newton's Polygon * Determination of the Genus * Holomorphic Differentials * The Riemann-Roch Theorem * The Genus is a Birational Invariant * PART 5: Miscellany * On the So-Called Fundamental Theorem of Algebra * Proof by Contradiction and the Sylow Theorems * Overview of 'Linear Algebra' * The Spectral Theorem * Kronecker as One of E.T. Bell's 'Men of Mathematics' * References
Harold M. Edwards is Emeritus Professor of Mathematics at New York University. His previous books are Advanced Calculus (1969, 1980, 1993), Riemann's Zeta Function (1974, 2001), Fermat's Last Theorem (1977), Galois Theory (1984), Divisor Theory (1990) and Linear Algebra (1995). Readers of his Advanced Calculus will know that his preference for constructive mathematics is not new. In 1980 he was awarded the Steele Prize for mathematical exposition for the Riemann and Fermat books.
From the reviews: "Harold Edwards is well known for his books with a constructivist slant, and his latest book aims to spread his message further. … The major part of the book consists of essays … telling a connected story, showing what can be achieved with such a constructive restriction imposed. The achievement is impressive … . " (John Baylis, The Mathematical Gazette, Vol. 90 (5l9), 2006) "It is not a book about the history/philosophy of mathematics but rather a very serious book of mathematics. … the mathematics is accessible to those with advanced undergraduate or graduate level courses in algebra … . Without a doubt the mathematics in this book is rigorous … . One of the nice features of this book is the bibliography which notes the sections where each reference appears. It should appeal to mathematicians and historians of mathematics alike." (Bonnie Shulman, MathDL, January, 2005) "The author of this volume points out immediately that it is not about the history or philosophy of mathematics, but rather a book about mathematics. It soon becomes clear, though, that historical and philosophical issues strongly influenced the topics discussed. … The general point of view is that all definitions, theorems, constructs, and proofs should involve only algorithms that terminate in a finite number of steps. … The book contains a wealth of interesting mathematics well worth reading." (Larry C. Grove, SIAM Review, Vol. 47 (4), 2005) “The book under review presents several important topics in mathematics from a constructivist point of view. … This book is a delight to read. … Moreover, the required background is kept to a minimum, so the book can be read by anyone with a good understanding of basic algebra. … The style of the book itself is sufficient reason for reading it. … One would wish that more mathematicians read the masters in search of inspiration, instead of merely following the most recent fads.” (S. C. Coutinho, SIGACT News, Vol. 41 (2), 2010)
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