Essays in Constructive Mathematics [Paperback]

Contents and treatment are fresh and very different from the standard treatments

Presents a fully constructive version of what it means to do algebra

The exposition is not only clear, it is friendly, philosophical, and considerate even to the most naive or inexperienced reader

He [Kronecker] was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number theory and alge? braic geometry as special cases.Andre Weil [62] This book is about mathematics, not the history or philosophy of mathemat? ics. Still, history and philosophy were prominent among my motives for writing it, and historical and philosophical issues will be major factors in determining whether it wins acceptance. Most mathematicians prefer constructive methods. Given two proofs of the same statement, one constructive and the other not, most will prefer the constructive proof. The real philosophical disagreement over the role of con? structions in mathematics is between thosethe majoritywho believe that to exclude from mathematics all statements that cannot be proved construc? tively would omit far too much, and those of us who believe, on the contrary, that the most interesting parts of mathematics can be dealt with construc? tively, and that the greater rigor and precision of mathematics done in that way adds immensely to its value.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} + ..lÓ$