This treatise covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields, and finitely generated fields. The author includes several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields. Some of these results are based on the recent work of E. Bombieri and U. Zannier, presented here by Zannier in an appendix. The book also treats other subjects such as Ritt's theory of composition of polynomials, and properties of the Mahler measure and concludes with a bibliography of over 300 items
Author(s): Barr M., Wells C.
Publisher: PH
Year: 1990
Language: English
Pages: 448
Tags: Математика;Общая алгебра;Теория категорий;
Title......Page 3
Contents......Page 5
Preface......Page 9
1.1 Sets......Page 17
1.2 Functions......Page 18
1.3 Graphs......Page 23
1.4 Homomorphisms of graphs......Page 25
2.1 Basic definitions......Page 29
2.2 Functional programming languages as categories......Page 33
2.3 Mathematical structures as categories......Page 37
2.4 Categories of sets with structure......Page 40
2.5 Categories of algebraic structures......Page 44
2.6 Constructions on categories......Page 47
2.7 Properties of objects and arrows in a category......Page 52
2.8 Monomorphisms and subobjects......Page 57
2.9 Other types of arrow......Page 61
3.1 Functors......Page 67
3.2 Actions......Page 75
3.3 Types of functors......Page 80
3.4 Equivalences......Page 83
3.5 Quotient categories......Page 87
4.1 Diagrams......Page 92
4.2 Natural transformations......Page 99
4.3 Natural transformations between functors......Page 104
4.4 The Godement calculus of natural transformations......Page 110
4.5 The Yoneda Lemma and universal elements......Page 113
4.6 Linear sketches (graphs with diagrams)......Page 119
4.7 Linear sketches with constants: initial term models......Page 123
5.1 The product of two objects in a category......Page 130
5.2 Notation for and properties of products......Page 134
5.3 Finite products......Page 143
5.4 Sums......Page 149
5.5 Natural numbers objects......Page 152
5.6 Deduction systems as categories......Page 156
6.1 Cartesian closed categories......Page 158
6.2 Typed A-calculus......Page 166
6.3 ?-calculus to category and back )......Page 168
6.4 Arrows vs. terms......Page 170
6.5 Fixed points in cartesian closed categories......Page 172
7 Finite discrete sketches......Page 177
7.1 Finite product sketches......Page 178
7.2 The sketch for semigroups......Page 183
7.3 Notation for FP sketches......Page 188
7.4 Arrows between models of FP sketches......Page 191
7.5 The theory of an FP sketch......Page 193
7.6 Initial term models for FP sketches......Page 195
7.7 Signatures and FP sketches......Page 199
7.8 FD sketches......Page 203
7.9 The sketch for fields......Page 205
7.10 Term algebras for FD sketches......Page 208
8.1 Equalizers......Page 216
8.2 The general concept of limit......Page 219
8.3 Pullbacks......Page 223
8.4 Coequalizers......Page 228
8.5 Cocones......Page 230
8.6 More about sums......Page 235
8.7 Unification as coequalizer......Page 239
9.1 Finite limit sketches......Page 244
9.2 Initial term models of FL sketches......Page 249
9.3 The theory of an FL sketch......Page 252
9.4 General definition of sketch......Page 253
10.1 Homomorphisms of sketches......Page 257
10.2 Parametrized data types as pushouts......Page 258
10.3 The model category functor......Page 264
11.1 Fibrations......Page 268
11.2 The Grothendieck construction......Page 273
11.3 An equivalence of categories......Page 279
11.4 Wreath products......Page 282
12.1 Free monoids......Page 287
12.2 Adjoints......Page 290
12.3 Further topics on adjoints......Page 297
12.4 Locally cartesian closed categories......Page 301
13.1 Fixed points for a functor......Page 304
13.2 Recursive categories......Page 309
13.3 Triples......Page 313
13.4 Factorizations of a triple......Page 314
13.5 Scott domains......Page 318
14 Toposes......Page 324
14.1 Definition of topos......Page 325
14.2 Properties of toposes......Page 328
14.3 Is a two-element poset complete?......Page 332
14.4 Presheaves......Page 334
14.5 Sheaves......Page 336
14.6 Fuzzy sets......Page 341
14.7 External functors......Page 344
14.8 The realizability topos......Page 349
Appendix: Solutions to exercises......Page 353
Bibliography......Page 433
Index......Page 443
