Essays in constructive mathematics
著者
書誌事項
Essays in constructive mathematics
Springer, c2005
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注記
Includes bibliographical references (p. [205]-208) and index
内容説明・目次
内容説明
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
目次
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
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