Undergraduate AlgebraThis 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. |
Contents
CHAPTER | 1 |
2 Basic Properties 2572 | 2 |
3 Greatest Common Divisor | 6 |
4 Unique Factorization | 11 |
5 Equivalence Relations and Congruences | 15 |
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 III | 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 |
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 SL C Iwasawa Decompositions | 245 |
5 Other Decompositions | 252 |
6 The Conjugation Action | 254 |
CHAPTER VII | 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 | 302 |
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 |
Appendix | 372 |
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Common terms and phrases
a₁ abelian group additive group algebraically closed assume automorphism bijective called card(A Cauchy sequence Chapter coefficients commutative ring complex numbers conjugation Corollary coset cosets of H cyclic group define denote denumerable direct sum divides element of G embedding Example Exercise extension of F finite extension finite field finite group follows functions G₁ Galois extension Galois group group G group of order H₁ Hence induction injective integer inverse irreducible polynomial isomorphism kernel left ideal Lemma Let f Let f(t Let G Let H linear map m₁ matrices module multiplicative group n-th root non-zero normal subgroup notation number of elements permutation positive integer prime number properties rational numbers real numbers relatively prime ring-homomorphism root of unity Show solvable splitting field subgroup of G subset subspace Suppose surjective Theorem uniquely determined unit element v₁ vector space verify whence write xe G Z/nZ