Algebra: Volume I: Fields and Galois Theory

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Springer Science & Business Media, Jul 2, 2006 - Mathematics - 296 pages
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From Math Reviews: "This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the framework of the development of carefully selected problems. Thus, for instance, the transformation of the classical geometrical problems on constructions with ruler and compass in their algebraic setting in the first chapter introduces the reader spontaneously to such fundamental algebraic notions as field extension, the degree of an extension, etc... The book ends with an appendix containing exercises and notes on the previous parts of the book. However, brief historical comments and suggestions for further reading are also scattered through the text."

 

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Contents

Foreword
3
Algebraic Extensions
15
Simple Extensions
21
Fundamentals of Divisibility
33
Prime Factorization in Polynomial Rings Gausss Theorem 45
44
Polynomial Splitting Fields
55
Separable Extensions
65
Galois Extensions 75
74
Norm and Trace
133
Binomial Equations 143
142
Solvability of Equations
165
Integral Ring Extensions
191
The Transcendence of
203
Transcendental Field Extensions
209
Hilberts Nullstellensatz
217
Problems and Remarks
231

Finite Fields Cyclic Groups and Roots of Unity
83
Group Actions
93
Applications of Galois Theory to Cyclotomic Fields
103
Further Steps into Galois Theory 115
114

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