Algebra

by Lorenz, Falko

• ISBN13: 9780387289304
• ISBN10: 0387289305
• Format: Paperback
• Copyright: 2005-12-01
• Publisher: Springer Nature

Summary

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."

Table of Contents

Foreword	v
1 Constructibility with Ruler and Compass	1	(14)
2 Algebraic Extensions	15	(6)
3 Simple Extensions	21	(12)
4 Fundamentals of Divisibility	33	(12)
5 Prime Factorization in Polynomial Rings. Gauss's Theorem	45	(10)
6 Polynomial Splitting Fields	55	(10)
7 Separable Extensions	65	(10)
8 Galois Extensions	75	(8)
9 Finite Fields, Cyclic Groups and Roots of Unity	83	(10)
10 Group Actions	93	(10)
11 Applications of Galois Theory to Cyclotomic Fields	103	(12)
12 Further Steps into Galois Theory	115	(18)
13 Norm and Trace	133	(10)
14 Binomial Equations	143	(22)
15 Solvability of Equations	165	(26)
16 Integral Ring Extensions	191	(12)
17 The Transcendence of π	203	(6)
18 Transcendental Field Extensions	209	(8)
19 Hilbert's Nullstellensatz	217	(14)
Appendix: Problems and Remarks	231	(52)
Index of Notation	283	(4)
Index	287