Kaplansky I. Fields and Rings 1969

Textbook in PDF and DJVU formats

These lecture notes combine three items previously available from Chicago's Department of Mathematics: Theory of Fields, Notes on Ring Theory, and Homological Dimension of Rings and Modules. I hope the material will be useful to the mathematical community and more convenient in the new format. A number of minor changes have been made; these are described in the introductions that precede the three sections. One point should be noted: the theorems are numbered consecutively within each section. Since there are no crossreferences between the sections, no confusion should result. I trust the reader will not mind a lack of complete consistency, e.g. in Part Il the modules are right and the mappings are placed on the right, while in Part III both get switched to the left.
Fields
Introduction
Field extensions
Ruler and compass constructions
Foundations of Galois theory
Normality and stability
Splitting fields
Radical extensions
The trace and norm theorems
Finite fields
Simple extensions
Cubic and quartic equations
Separability
Miscellaneous results on radical extensions
Infinite algebraic extensions
Rings
Introduction
The radical
Primitive rings and the density theorem
Semi-simple rings
The Wedderburn principal theorem
Theorems of Hopkins and Levitzki
Primitive rings with minimal ideals and dual vector spaces
Simple rings
Homological Dimension
Introduction
Dimension of modules
Global dimension
First theorem on change of rings
Polynomial rings
Second theorem on change of rings
Third theorem on change of rings
Localization
Preliminary lemmas
A regular ring has finite global dimension
A local ring of finite global dimension is regular
Injective modules
The group of homomorphisms
The vanishing of Ext
Injective dimension