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."
Author(s): Falko Lorenz, Silvio Levy
Series: Universitext
Edition: 1
Publisher: Springer
Year: 2005
Language: English
Pages: 292
Algebra Volume I: Fields and Galois Theory......Page 1
Contents......Page 7
01 - Constructibility with Ruler and Compass......Page 8
02 - Algebraic Extensions......Page 21
03 - Simple Extensions......Page 27
04 - Fundamentals of Divisibility......Page 39
05 - Prime Factorization in Polynomial Rings. Gauss’s Theorem......Page 50
06 - Polynomial Splitting Fields......Page 59
07 - Separable Extensions......Page 69
08 - Galois Extensions......Page 78
09 - Finite Fields, Cyclic Groups and Roots of Unity......Page 86
10 - Group Actions......Page 96
11 - Applications of Galois Theory to Cyclotomic Fields......Page 105
12 - Further Steps into Galois Theory......Page 116
13 - Norm and Trace......Page 133
14 - Binomial Equations......Page 142
15 - Solvability of Equations......Page 164
16 - Integral Ring Extensions with Applications to Galois Theory......Page 190
17 - The Transcendence of π......Page 202
19 - Hilbert’s Nullstellensatz......Page 216
Appendix: Problems and Remarks......Page 230
Index of Notation......Page 282
Index......Page 286
