Algebra

1. Front Matter

   Pages i-x

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2. Groups

   • William A. Adkins, Steven H. Weintraub

   Pages 1-48

3. Rings

   • William A. Adkins, Steven H. Weintraub

   Pages 49-106

4. Modules and Vector Spaces

   • William A. Adkins, Steven H. Weintraub

   Pages 107-181

5. Linear Algebra

   • William A. Adkins, Steven H. Weintraub

   Pages 182-288

6. Matrices over PIDs

   • William A. Adkins, Steven H. Weintraub

   Pages 289-340

7. Bilinear and Quadratic Forms

   • William A. Adkins, Steven H. Weintraub

   Pages 341-394

8. Topics in Module Theory

   • William A. Adkins, Steven H. Weintraub

   Pages 395-437

9. Group Representations

   • William A. Adkins, Steven H. Weintraub

   Pages 438-506

10. Back Matter

    Pages 507-529

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This book is designed as a text for a first-year graduate algebra course. As necessary background we would consider a good undergraduate linear algebra course. An undergraduate abstract algebra course, while helpful, is not necessary (and so an adventurous undergraduate might learn some algebra from this book). Perhaps the principal distinguishing feature of this book is its point of view. Many textbooks tend to be encyclopedic. We have tried to write one that is thematic, with a consistent point of view. The theme, as indicated by our title, is that of modules (though our intention has not been to write a textbook purely on module theory). We begin with some group and ring theory, to set the stage, and then, in the heart of the book, develop module theory. Having developed it, we present some of its applications: canonical forms for linear transformations, bilinear forms, and group representations. Why modules? The answer is that they are a basic unifying concept in mathematics. The reader is probably already familiar with the basic role that vector spaces play in mathematics, and modules are a generaliza­ tion of vector spaces. (To be precise, modules are to rings as vector spaces are to fields.

• Permutation
• algebra
• automorphism
• field
• homomorphism
• linear algebra
• matrices
• matrix
• quadratic form
• transformation

• Department of Mathematics, Louisiana State University, Baton Rouge, USA

  William A. Adkins, Steven H. Weintraub

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