Essays in Constructive Mathematics

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, Gausss theory of binary quadratic forms and Abels theorem about integrals of rational differentials on algebraic curves. For Abels theorem the main algorithm is Newtons 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.