AlgebraThis book comes from the first part of the lecture notes which the author used for a first-year graduate algebra course. The aim of this book is not only to give the students quick access to the basic knowledge of algebra, either for future advancement in the field of algebra, or for general background information, but also to show that algebra is truly a master key or a ?skeleton key? to many mathematical problems. As one knows, the teeth of an ordinary key prevent it from opening all but one door; whereas the skeleton key keeps only the essential parts, allow it to unlock many doors. The author wishes to present this book as an attempt to re-establish the contacts between algebra and other branches of mathematics and sciences. Many examples and exercises are included to illustrate the power of intuitive approaches to algebra. |
Contents
Chapter Set theory and Number Theory | 1 |
2 Unique Factorization Theorem | 6 |
3 Congruence | 12 |
4 Chinese Remainder Theorem | 20 |
5 Complex Integers | 23 |
6 Real Numbers and padic Numbers | 33 |
Group theory | 47 |
2 The Transformation Groups on Sets | 55 |
3 Linear Transformation and Matrix | 181 |
4 Module and Module over P I D | 196 |
5 Jordan Canonical Form | 214 |
6 Characteristic Polynomial | 223 |
7 Inner Product and Bilinear form | 232 |
8 Spectral Theory | 243 |
Polynomials in One Variable and Field Theory | 252 |
2 Algebraic Extension | 257 |
3 Subgroups | 62 |
4 Normal Subgroups and Inner Automorphisms | 73 |
5 Automorphism Groups | 82 |
6 pGroups and Sylow Theorems | 85 |
7 JordanHölder Theorem | 89 |
8 Symmetric Group Sn | 96 |
Polynomials | 102 |
2 Polynomial Rings and Quotient Fields | 108 |
3 Unique Factorization Theorem for Polynomials | 114 |
4 Symmetric Polynomial Resultant and Discriminant | 130 |
5 Ideals | 144 |
Linear Algebra | 160 |
2 Basis and Dimension | 165 |
3 Algebraic Closure | 271 |
4 Characteristic and Finite Field | 274 |
5 Separable Algebraic Extension | 282 |
6 Galois Theory | 291 |
vii | 297 |
7 Solve Equation by Radicals | 306 |
8 Field Polynomial and Field Discriminant | 321 |
9 Lüroths Theorem | 326 |
Appendix | 332 |
A2 Peanos Axioms | 333 |
A3 Homological Algebra | 337 |
| 343 | |
Common terms and phrases
a₁ algebraic closure algebraic extension assume b₁ B₂ bijective c₁ canonical map characteristic polynomial characteristic values claim coefficients common divisor commutative group commutative ring complex integers complex numbers conclude constructed Corollary deg f(x Discussion easy Example factor finite extension finite field FL(n following definition following equation following theorem follows from Theorem Galois extension group G group of order homomorphism ideal inner product integral domain invertible irreducible polynomial isomorphism Jordan canonical form Let f(x Let G Let us consider Let us define linear transformation linearly independent set m₁ matrix form minimal polynomial module multiple root N₁ N₂ non-zero normal subgroup Note orbits orthonormal basis polynomial f(x positive integer prime number primitive polynomial Proof Prove quotient groups R-module real numbers root of unity Show splitting field subgroup of G submodule subset subspace Suppose surjective torsion transformation group u₁ unique vector space w₁