A Course in Constructive AlgebraThe constructive approach to mathematics has enjoyed a renaissance, caused in large part by the appearance of Errett Bishop's book Foundations of constr"uctiue analysis in 1967, and by the subtle influences of the proliferation of powerful computers. Bishop demonstrated that pure mathematics can be developed from a constructive point of view while maintaining a continuity with classical terminology and spirit; much more of classical mathematics was preserved than had been thought possible, and no classically false theorems resulted, as had been the case in other constructive schools such as intuitionism and Russian constructivism. The computers created a widespread awareness of the intuitive notion of an effecti ve procedure, and of computation in principle, in addi tion to stimulating the study of constructive algebra for actual implementation, and from the point of view of recursive function theory. In analysis, constructive problems arise instantly because we must start with the real numbers, and there is no finite procedure for deciding whether two given real numbers are equal or not (the real numbers are not discrete) . The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der waerden, made important contributions to construc tive algebra. Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden'of analysis rather than a theory of discrete algebraic structures. |
Contents
Chapter I Sets | 1 |
2 SETS SUBSETS AND FUNCTIONS | 7 |
3 CHOICE | 14 |
4 CATEGORIES | 16 |
5 PARTIALLY ORDERED SETS AND LATTICES | 20 |
6 WELLPOUNDED SETS AND ORDINALS | 24 |
NOTES | 30 |
Chapter II Basic Algebra | 35 |
4 THE FUNDAMENTAL THEOREM OF ALGEBRA | 189 |
NOTES | 192 |
Chapter VIII Commutative Noetherian Rings | 193 |
2 NOETHER NORMALIZATION AND THE ARTINREES LEMMA | 197 |
3 THE NULLSTELLENSATZ | 201 |
4 TENNENBAUMS APPROACH TO THE HILBERT BASIS THEOREM | 204 |
5 PRIMARY IDEALS | 208 |
6 LOCALIZATION | 211 |
2 RINGS AND FIELDS | 41 |
3 REAL NUMBERS | 48 |
4 MODULES | 52 |
5 POLYNOMIAL RINGS | 60 |
6 MATRICES AND VECTOR SPACES | 65 |
7 DETERMINANTS | 69 |
8 SYMMETRIC POLYNOMIALS | 73 |
NOTES | 77 |
Chapter III Rings and Modules | 78 |
2 COHERENT AND NOETHERIAN NODULES | 80 |
3 LOCALIZATION | 85 |
4 TENSOR PRODUCTS | 88 |
5 FLAT MODULES | 92 |
6 LOCAL RINGS | 96 |
7 COMMUTATIVE LOCAL RINGS | 102 |
NOTES | 107 |
Chapter IV Divisibility in Discrete Domains | 108 |
2 UFDS AND BEZOUT DOMAINS | 114 |
3 DEDEKINDHASSE RINGS AND EUCLIDEAN DOMAINS | 117 |
4 POLYNOMIAL RINGS | 123 |
NOTES | 126 |
Chapter V Principal Ideal Domains | 128 |
2 FINITELY PRESENTED NODULES | 130 |
3 TORSION MODULES pCOMPONENTS ELEMENTARY DIVISORS | 133 |
4 LINEAR TRANSFORMATIONS | 135 |
NOTES | 138 |
Chapter VI Field Theory | 139 |
2 ALGEBRAIC INDEPENDENCE AND TRANSCENDENCE BASES | 145 |
3 SPLITTING FIELDS AND ALGEBRAIC CLOSURES | 150 |
4 SEPARABILITY AND DIAGONALIZABILITY | 154 |
5 PRIMITIVE ELEMENTS | 158 |
6 SEPARABILITY AND CHARACTERISTIC p | 161 |
7 PERFECT FIELDS | 164 |
8 GALOIS THEORY | 167 |
NOTES | 175 |
Chapter VII Factoring Polynomials | 176 |
2 EXTENSIONS OF SEPARABLY FACTORIAL FIELDS | 182 |
3 SEIOENBERG FIELDS | 186 |
7 PRIMARY DECOMPOSITIONS | 216 |
8 LASKERNOETHER RINGS | 220 |
9 FULLY LASKERNOETHER RINGS | 224 |
10 THE PRINCIPAL IDEAL THEOREM | 228 |
NOTES | 231 |
Chapter IX Finite Dimensional Algebras | 232 |
2 THE DENSITY THEOREM | 235 |
3 THE RADICAL AND SUMMANDS | 237 |
4 WEDDERBURNS THEOREM PART ONE | 242 |
5 MATRIX RINGS AND DIVISION ALGEBRAS | 245 |
NOTES | 248 |
Chapter X Free Groups | 249 |
2 NIELSEN SETS | 253 |
3 FINITELY GENERATED SUBGROUPS OF FREE GROUPS | 255 |
4 DETACHABLE SUBGROUPS OF FINITERANK FREE GROUPS | 257 |
5 CONJUGATE SUBGROUPS | 261 |
NOTES | 263 |
Chapter XI Abelian Groups | 265 |
2 DIVISIBLE GROUPS | 269 |
3 HEIGHT FUNCTIONS ON pGROUPS | 273 |
4 ULNS THEOREM | 277 |
5 CONSTRUCTION OF ULM GROUPS | 281 |
NOTES | 285 |
Chapter XII Valuation Theory | 287 |
2 LOCALLY PRECOMPACT VALUATIONS | 292 |
3 PSEUDOFACTORIAL FIELDS | 295 |
4 NORMED VECTOR SPACES | 299 |
5 REAL AND COMPLEX FIELDS | 302 |
6 HENSELS LEMMA | 306 |
7 EXTENSIONS OF VALUATIONS | 315 |
8 e AND f | 319 |
NOTES | 324 |
Chapter XIII Dedekind Domains | 326 |
2 IDEAL THEORY | 329 |
3 FINITE EXTENSIONS | 332 |
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Common terms and phrases
abelian group algebraic algebraically dependent algorithm assume binary sequence Brouwerian example coefficients commutative ring Construct a Brouwerian containing COROLLARY cyclic defined detachable ideals dimensional direct sum discrete field division ring endomorphism Exercise exists extension factorial field field of quotients finite number finite set finite subset finite-rank free finitely generated ideal finitely generated prime finitely generated submodule finitely presented follows free group free module function Henselian Heyting field homomorphism implies induction inequality invertible irreducible isomorphic kernel Lasker-Noether ring left ideal left R-module LEMMA Let f local ring matrix monic polynomial monoid multiplicative submonoid nilpotent Noetherian ring nonarchimedean nontrivial nonzero element p-group polynomial f polynomial in k[X positive integer primary ideal principal ideal domain PROOF R-module rational numbers real numbers root satisfies separable polynomial separably factorial splitting field strongly relatively prime subfield subgroup submodule summand Suppose THEOREM torsion-free unique valuation vector space