Springer Science & Business Media, Oct 31, 2006 - 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 II 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 m 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 7 165 10 The abc Conjecture 171 CHAPTER V 177 2 Dimension of a Vector Space 185
 Some Linear Groups 232 2 Structure of GL2F 236 3 SL2F 239 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 CHAPTER VIII 306 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 Copyright