Undergraduate Algebra

Front Cover
Springer Science & Business Media, Mar 21, 2005 - Mathematics - 389 pages
This book, together with Linear Algebra, constitutes a curriculum for an algebra program addressed to undergraduates. The separation of the hnear algebra from the other basic algebraic structures fits all existing tendencies affecting undergraduate teaching, and I agree with these tendencies. I have made the present book self contained logically, but it is probably better if students take the linear algebra course before being introduced to the more abstract notions of groups, rings, and fields, and the systematic development of their basic abstract properties. There is of course a little overlap with the book Lin ear Algebra, since I wanted to make the present book self contained. I define vector spaces, matrices, and linear maps and prove their basic properties. The present book could be used for a one-term course, or a year's course, possibly combining it with Linear Algebra. I think it is important to do the field theory and the Galois theory, more important, say, than to do much more group theory than we have done here. There is a chapter on finite fields, which exhibit both features from general field theory, and special features due to characteristic p. Such fields have become important in coding theory.
 

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Contents

CHAPTER
1
2 Basic Properties
2
3 Greatest Common Divisor
5
4 Unique Factorization
7
5 Equivalence Relations and Congruences
12
CHAPTER ll
16
2 Mappings
26
3 Homomorphisms
33
3 Matrices and Linear Maps
188
4 Modules
192
5 Factor Modules
203
6 Free Abelian Groups
205
7 Modules over Principal Rings
210
8 Eigenvectors and Eigenvalues
214
9 Polynomials of Matrices and Linear Maps
220
CHAPTER VI
231

4 Cosets and Normal Subgroups
41
5 Application to Cyclic Groups
55
6 Permutation Groups
59
7 Finite Abelian Groups
67
8 Operation of a Group on a Set
73
9 Sylow Subgroups
79
CHAPTER HI Rings
83
2 Ideals
87
3 Homomorphisms
90
4 Quotient Fields
100
Polynomials
105
2 Greatest Common Divisor
118
3 Unique Factorization
120
4 Partial Fractions
129
5 Polynomials Over Rings and Over the Integers
136
6 Principal Rings and Factorial Rings
143
7 Polynomials in Several Variables
152
8 Symmetric Polynomials
159
9 The MasonStothers Theorem
165
l0 The abc Conjecture
171
CHAPTER V
177
2 Dimension of a Vector Space
185
Some Linear Groups
232
2 Structure of GL2F
236
3 sLF
241
4 SLR and SLC Iwasawa Decompositions
245
5 Other Decompositions
252
6 The Conjugation Action
254
CHAPTER Vll Field Theory
258
2 Embeddings
267
3 Splitting Fields
275
4 Galois Theory
280
5 Quadratic and Cubic Extensions
292
6 Solvability by Radicals
296
7 Infinite Extensions
305
Finite Fields
309
2 The Frobenius Automorphism
313
3 The Primitive Elements
315
2 Preliminaries
330
4 Decimal Expansions
343
3 Cardinal Numbers
359
Appendh
372
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