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Algebraic Extensions of Fields

Paul J. McCarthy
Publisher: 
Dover Publications
Publication Date: 
1991
Number of Pages: 
166
Format: 
Paperback
Price: 
8.95
ISBN: 
978-0486666518
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

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CHAPTER 1. Algebraic Extensions
1. Definitions
2. Algebraic extensions
3. Characteristic: perfect fields
4. Separability of extensions
5. Normal extensions
6. Finite fields
7. Primitive elements
8. Algebraically closed fields
9. Norms and traces
EXERCISES
CHAPTER 2. Galois Theory
1. Automorphisms of extensions: Galois extensions
2. The fundamental theorem of Galois theory
3. An example
4. Cyclotomic fields
5. The first cohomolgy group
6. Cyclic extensions
7. Multiplicative Kummer theory
8. Additive Kummer theory
9. Solutions of polynomial equations by radicals
10. Infinite Galois extensions
11. The Krull topology
12. Inverse limits
EXERCISES
CHAPTER 3. Introduction to Valuation Theory
1. Definition of valuation: examples
2. Valuations on the fields Q and k(x)
3. Complete fields and completions
4. Value groups and residue class fields
5. Prolongations of valuations
6. Relatively complete fields
7. "Prolongations of valuations, continued"
EXERCISES
CHAPTER 4. Extensions of Valuated Fields
1. Ramification and residue class degree
2. Unramified and tamely ramified extensions
3. The different
4. Extensions K/k with K/k separable
5. Ramification groups
EXERCISES
CHAPTER 5. Dedekind Fields
1. The fundamental theorem of Dedekind fields
2. Extensions of Dedekind fields
3. Factoring of ideals in extensions
4. Galois extensions of Dedekind fields
EXERCISES
APPENDIX 1. Proof of Theorem 19 of Chapter 2
APPENDIX 2. Example of the Galois Group of an Infinite Extension
BIBLIOGRAPHY
INDEX

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