Lectures on abstract algebra

707 Pages·2016·2.62 MB·English

by  Elman R

Lectures on Abstract Algebra
Preliminary Version
Richard Elman
Department of Mathematics,
University of California,
Los Angeles, CA 90095-1555, USA

Contents
Part 1. Preliminaries 1
1. Introduction 3
Chapter I. The Integers 11
2. Well-Ordering and Induction 11
3. Addendum: The Greatest Integer Function 16
4. Division and the Greatest Common Divisor 20
Chapter II. Equivalence Relations 29
5. Equivalence Relations 29
6. Modular Arithmetic 33
7. Surjective maps 39
Part 2. Group Theory 43
Chapter III. Groups 45
8. Definitions and Examples 45
9. First Properties 54
10. Cosets 59
11. Homomorphisms 64
12. The First Isomorphism Theorem 70
13. The Correspondence Principle 75
14. Finite Abelian Groups 79
15. Addendum: Finitely Generated Groups 83
16. Series 86
17. Free Groups 93
Chapter IV. Group Actions 101
18. The Orbit Decomposition Theorem 101
19. Addendum: Finite Rotation Groups in R3. 105
20. Examples of Group Actions 108
21. Sylow Theorems 118
22. The Symmetric and Alternating Groups 126
Part 3. Ring Theory 139
iii
iv CONTENTS
Chapter V. General Properties of Rings 141
23. Definitions and Examples 141
24. Factor Rings and Rings of Quotients 153
25. Zorn’s Lemma 162
26. Addendum: Localization 170
Chapter VI. Domains 173
27. Special Domains 173
28. Addendum: Characterization of UFDs 181
29. Gaussian Integers 182
30. Addendum: The Four Square Theorem 189
Chapter VII. Polynomial Rings 195
31. Introduction to Polynomial Rings 195
32. Polynomial Rings over a UFD 202
33. Addendum: Polynomial Rings over a Field 208
34. Addendum: Algebraic Weierstraß Preparation Theorem 210
Part 4. Module Theory 217
Chapter VIII. Modules 219
35. Basic Properties of Modules 219
36. Free Modules 231
Chapter IX. Noetherian Rings and Modules 241
37. Noetherian Modules 241
38. Hilbert’s Theorems 245
39. Addendum: Affine Plane Curves 252
Chapter X. Finitely Generated Modules Over a PID 257
40. Smith Normal Form 257
41. The Fundamental Theorem 263
42. Canonical Forms for Matrices 277
43. Addendum: Jordan Decomposition 290
44. Addendum: Cayley-Hamilton Theorem 293
Part 5. Field Theory 297
Chapter XI. Field Extensions 299
45. Algebraic Elements 299
46. Addendum: Transcendental Extensions 310
47. Splitting Fields 312
48. Algebraically Closed Fields 323
49. Constructible Numbers 326
CONTENTS v
50. Separable Elements 337
Chapter XII. Galois Theory 343
51. Characters 343
52. Computations 350
53. Galois Extensions 356
54. The Fundamental Theorem of Galois Theory 365
55. Addendum: Infinite Galois Theory 374
56. Roots of Unity 378
57. Radical Extensions 391
58. Addendum: Kummer Theory 405
59. Addendum: Galois’ Theorem 408
60. The Discriminant of a Polynomial 412
61. Purely Transcendental Extensions 416
62. Finite Fields 419
63. Addendum: Hilbert Irreducibility Theorem 424
Part 6. Additional Topics 439
Chapter XIII. Transcendental Numbers 441
64. Liouville Numbers 441
65. Transcendence of e 444
66. Symmetric Functions 446
67. Transcendence of π 449
Chapter XIV. The Theory of Formally Real Fields 457
68. Orderings 457
69. Extensions of Ordered Fields 460
70. Characterization of Real Closed Fields 467
71. Hilbert’s 17th Problem 470
Chapter XV. Dedekind Domains 477
72. Integral Elements 477
73. Integral Extensions of Domains 481
74. Dedekind Domains 484
75. Extension of Dedekind Domains 493
76. Hilbert Ramification Theory 498
77. The Discriminant of a Number Field 502
78. Dedekind’s Theorem on Ramification 507
79. The Quadratic Case 510
80. Addendum: Valuation Rings and Pru¨fer Domains 515
Chapter XVI. Introduction to Commutative Algebra 525
81. Zariski Topology 525
vi CONTENTS
82. Integral Extensions of Commutative Rings 536
83. Primary Decomposition 546
84. Akizuki and Krull-Akizuki Theorems 555
85. Affine Algebras 563
86. Addendum: Japanese Rings 576
Chapter XVII. Division and Semisimple Rings 579
87. Wedderburn Theory 579
88. The Artin-Wedderburn Theorem 588
89. Finite Dimensional Real Division Algebras 592
90. Addendum: Cyclic Algebras 594
91. Addendum: Jacobson’s Theorem 599
Chapter XVIII. Introduction to Representation Theory 603
92. Representations 603
93. Split Group Rings 610
94. Addendum: Hurwitz’s Theorem 613
95. Characters 617
96. Orthogonality Relations 620
97. Burnside’s paqb Theorem 626
98. Addendum: Torsion Linear Groups 630
Chapter XIX. Universal Properties and Multilinear Algebra 637
99. Some Universal Properties of Modules 638
100. Tensor Products 643
101. Tensor, Symmetric, and Exterior Algebras 650
102. The Determinant 656
Appendices 663
103. Matrix Representations 665
104. Smith Normal Form over a Euclidean Ring 669
105. Primitive Roots 672
106. Pell’s Equation 674
Bibliography 677
Notation 679
Index 683
Part 1
Preliminaries

1. INTRODUCTION 3
1. Introduction
In this introduction, we introduce some of the notations and defi-
nitions that we shall use throughout by means of investigating a few
mathematical statements. Consider the following statements:
Statements 1.1.
(1) Theinteger243112609−1, whichhas12978189digits, isaprime.
(2) There exist infinitely many (rational) primes.
(3) Let
π(x) := the number of positive primes ≤ x,
then
π(x)
lim = 1.
x→∞ x/logx
√
(4) 2 ∈/ Q, where Q is the set of rational numbers.
(5) The real number π is not “algebraic” over Q.
(6) There exist infinitely many real numbers [even “many more”
than the number of elements in Q] not algebraic over Q.
To look at these statements, we need some definitions. Let a,b ∈ Z,
where Z is the set of integers. We say that a divides b (in Z) if
b
there exists an integer n such that b = an,i.e., ∈ Z.
a
We write
a | b( in Z).
For example, 3 | 12 as 12 = 4 · 3 but 5(cid:54) | 12 in Z where (cid:54) | , means
does not divide.
An integer p is called a prime if p (cid:54)= 0,±1 and
n | p in Z implies that n ∈ {1,−1,p,−p}, i.e., n = ±1,±p.
For example, 2,3,7,11,... are prime. [The prime 2 is actually
the “oddest prime of all”!] We shall use that every integer n > 1 is
divisible by some prime (to be proven later). A complex number x is
called irrational if x is not a rational number, i.e., x ∈ C\Q := {z ∈
C | z ∈/ Q}, where C is the set of complex numbers.
A complex number x is called algebraic (over Q) if there exists
a nonzero polynomial f(t) = a tn + a tn−1 + ··· + a t + a with
n n−1 1 0
a ,...,a ∈ Q(somen)notallzerosatisfyingf(x) = a xn+a xn−1+
0 n n n−1
···+a x+a = 0. We shall write f for f(t) where t always represents
1 0
a variable and f(x) means plug x into f. We let
Q[t] := the set of polynomials with rational coefficients.
4
So x is algebraic over Q, if there exists 0 (cid:54)= f ∈ Q[t] such that f(x) = 0.
A complex number x that is not algebraic (over Q) is called transcen-
dental (over Q), so x ∈ C is transcendental if there exists no nonzero
polynomial f ∈ Q[t] satisfying f(x) = 0.
With the above definitions, we can look at our statements to see
which ones are interesting, deep, etc.
WestartwithStatement1above. ThiswasshownbyaUCLAteam,
usingaprimalitytestofLucasonMersennenumbers, tobeaprime, the
first known prime having ten million digits. It is, on the face of it, not
very interesting. After all, it is analogous to saying 97 is a prime. How-
ever, it is interesting historically. Call an integer M := 2n−1, n a pos-
n
itive integer, a Mersenne number and a Mersenne prime if it is a prime.
In 1644, Mersenne conjectured that for n ≤ 257, the Mersenne number
M is prime if and only if n = 2,3,5,7,13,17,19,31,67,127,257. [If
n
M is a prime, then n must be a prime as 2ab −1 factors if a,b > 1.]
n
It was shown in the 1880’s that M was a prime. In 1903 Frank Cole
61
showed that M = 193707721·761838257287. [He silently multiplied
67
out these two numbers on a blackboard at a meeting of the American
Mathematical Society.] In 1931 it was shown that M was compos-
257
ite, i.e., not 0,±1, or a prime, finally finishing the original Mersenne’s
conjecture with the correct solution. Mersenne was interested in these
numbers because of its connection to a study in antiquity.
An integer n > 1 is called perfect if
(cid:88) (cid:88)
n = d or 2n = d,
d|n d|n
0<d<n 0<d≤n
e.g., 6 = 1 + 2 + 3. The first equation says that n is the sum of its
aloquot divisors. A theorem in elementary number theory says:
Theorem 1.2. (Euclid/Euler) An even number N is perfect if and
only if N = 1p(p+1) with p a Mersenne prime (so N = 2n−1(2n −1)
2
with p = 2n −1).
It is still an open problem whether there exists infinitely many even
perfect numbers, equivalently, infinitely many Mersenne primes and an
open problem whether there exists any odd perfect numbers.
Statement 2 is very interesting and is due to Euclid. It may be the
first deep mathematical fact that one learns. The proof is quite simple.
If the result is false, let p ,...,p be a complete list of (positive) primes
1 n
and set N = p ···p +1. We use (cf. Exercise 4.17), which says:
1 n
If x | y and x | z in Z, then for all a,b ∈ Z, we have x | ay +bz,