A Course in Constructive Algebra

Preface
CHAPTER I. SETS
 1. Constructive vs. classical mathematics
 2. Sets, subsets and functions
 3. Choice
 4. categories
 5. Partially ordered sets and lattices
 6. Well-founded sets and ordinals
 Notes
CHAPTER II. BASIC ALGEBRA
 1. Groups
 2. Rings and fields
 3. Real numbers
 4. Modules
 5. Polynomial rings
 6. Matrices and vector spaces
 7. Determinants
 8. Symmetric polynomials
 Notes
CHAPTER III. RINGS AND IDEALS
 1. Quasi-regular ideals
 2. Coherent and Noetherian modules
 3. Localization
 4. Tensor products
 5. Flat modules
 6. Local rings
 7. Commutative local rings
 Notes
CHAPTER IV. DIVISIBILITY IN DISCRETE DOMAINS
 1. Cancellation monoids
 2. UFD\'s and Bezout domains
 3. Dedekind-Hasse rings and Euclidean domains
 4. Polynomial rings
 Notes
CHAPTER V. PRINCIPAL IDEAL DOMAINS
 1. Diagonalizing matrices
 2. Finitely presented modules
 3. Torsion modules, p-components, elementary divisors
 4. Linear transformations
 Notes
CHAPTER VI. FIELD THEORY
 1. Integral extensions and impotent rings
 2. Algebraic independence and transcendence bases
 3. Splitting fields and algebraic closures
 4. Separability and diagonalizability
 5. Primitive elements
 6. Separability and characteristic p
 7. Perfect fields
 8. Galois theory
 Notes
CHAPTER VII. FACTORING POLYNOMIALS
 1. Factorial and separably factorial fields
 2. Extensions of (separably) factorial fields
 3. Condition P
 4. The fundamental theorem of algebra
 Notes
CHAPTER VIII. COMMUTATIVE NOETHERIAN RINGS
 1. The Hilbert basis theorem
 2. Noether normalization and the Artin-Rees lemma
 3. The Nullstellensatz
 4. Tennenbaum\'s approach to the Hilbert basis theorem
 5. Primary ideals
 6. Localization
 7. Primary decomposition
 8. Lasker-Noether rings
 9. Fully Lasker-Noether rings
 10. The principal ideal theorem
 Notes
CHAPTER IX. FINITE DIMENSIONAL ALGEBRAS
 1. Representations
 2. The density theorem
 3. The radical and summands
 4. Wedderburn\'s theorem, part one
 5. Matrix rings and division algebras
 Notes
CHAPTER X. FREE GROUPS
 1. Existence and uniqueness
 2. Nielsen sets
 3. Finitely generated subgroups
 4. Detachable subgroups of finite-rank free groups
 5. Conjugate subgroups
 Notes
CHAPTER XI. ABELIAN GROUPS
 1. Finite-rank torsion-free groups
 2. Divisible groups
 3. Height functions on p-groups
 4. Ulm\'s theorem
 5. Construction of Ulm groups
 Notes
CHAPTER XII. VALUATION THEORY
 1. valuations
 2. Locally precompact valuations
 3. Pseudofactorial fields
 4. Normed vector spaces
 5. Real and complex fields
 6. Hensel\'s lemma
 7. Extensions of valuations
 8. e and f
 Notes
CHAPTER XIII. DEDEKIND DOMAINS
 1. Dedekind sets of valuations
 2. Ideal theory
 3. Finite extensions
BIBLIOGRAPHY
INDEX