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

This book promotes constructive mathematics not by defining it or formalizing it but by practicing it. This means that its definitions and proofs use finite algorithms, not `algorithms' that require surveying an infinite number of possibilities to determine whether a given condition is met. The topics covered derive from classic works of nineteenth century mathematics - among them Galois' theory of algebraic equations, Gauss's theory of binary quadratic forms and Abel's theorem about integrals of rational differentials on algebraic curves. For Abel's theorem the main algorithm is Newton's polygon, which is given a full treatment. Other topics covered include the fundamental theorem of algebra, the factorization of polynomials over an algebraic number field, and the spectral theorem for symmetric matrices.

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