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This book combines in one volume Irving Kaplansky's lecture notes on the theory of fields, ring theory, and homological dimensions of rings and modules.
Preface Pt. I: Fields 1: Field extensions 2: Ruler and compass constructions 3: Foundations of Galois theory 4: Normality and stability 5: Splitting fields 6: Radical extensions 7: The trace and norm theorems 8: Finite fields 9: Simple extensions 10: Cubic and quartic equations 11: Separability 12: Miscellaneous results on radical extensions 13: Infinite algebraic extensions Pt. II: Rings 1: The radical 2: Primitive rings and the density theorem 3: Semi-simple rings 4: The Wedderburn principal theorem 5: Theorems of Hopkins and Levitzki 6: Primitive rings with minimal ideals and dual vector spaces 7: Simple rings Pt. III: Homological Dimension 1: Dimension of modules 2: Global dimension 3: First theorem on change of rings 4: Polynomial rings 5: Second theorem on change of rings 6: Third theorem on change of rings 7: Localization 8: Preliminary lemmas 9: A regular ring has finite global dimension 10: A local ring of finite global dimension is regular 11: Injective modules 12: The group of homomorphisms 13: The vanishing of Ext 14: Injective dimension Notes Index
