Making transcendence transparent

Cover......Page 1
Title page......Page 2
Contents......Page 4
Preface: The Journey Ahead......Page 8
0.2 The problem of addition......Page 10
0.3 The problem of multiplication......Page 11
0.4 A search for missing numbers......Page 13
0.5 Is i a number?......Page 14
0.6 Transcending the algebraic numbers......Page 15
Postscript: Tools of the transcendence trade......Page 16
1.1 Turning a mountain of transcendence into a molehill......Page 18
1.2 One small rational for Liouville, one giant step for transcendence......Page 19
1.3 Our first transcendental number......Page 20
1.4 The proof of Liouville's Theorem......Page 22
Algebraic Excursion: Irreducible polynomials evaluated at rational values......Page 23
1.5 Liouville numbers......Page 27
1.6 The number 0.12345678910111213141516.........Page 29
1.7 Roth's Theorem: The ultimate Liouville result......Page 33
1.8 Life after Liouville......Page 34
2.1 Fourier's proof of Euler's slick result......Page 36
2.2 A first attempt at a proof......Page 38
2.3 The classic vanishing polynomial trick......Page 41
2.4 The first part of the proof of Theorem 2.2 - The elusive estimate......Page 43
2.5 The dramatic conclusion of the proof Theorem 2.2 - Arithmetic conquers all......Page 45
2.6 The transcendence of e......Page 46
2.7 Foreshadowing algebraic exponents - The irrationality of exp(sqrt(Pi)) and Pi......Page 49
3.1 Algebraic exponents through the looking glass-A wonderland of transcendence......Page 52
3.2 Heading towards Hermite - A partial result casts some foreshadowing......Page 54
Algebraic Excursion: Symmetric functions and conjugates......Page 61
3.4 A conjugate appetizer - The delicate but surprisingly non-special, special case......Page 67
3.5 The main dish-Serving up a spaghetti of symmetry......Page 71
3.6 Algebraic Independence-Freeing ourselves of unhealthy dependencies......Page 79
4.1 Giving e^z a complex by taking away its power series......Page 86
4.2 Throwing in our two bits as a warm-up to the irrationality of en......Page 87
4.3 Our first ill-fated attempt at a proof-A star-crossed relationship between the order of vanishing and the degree......Page 92
4.4 Polynomials in two variables and the power of i......Page 96
4.5 The ever-popular polynomial construction......Page 97
Algebraic Excursion: Solving a system of linear equations in integers......Page 101
4.6 At long last, a proof that exp(Pi) is irrational......Page 105
4.7 A first glance at the transcendence of exp(Pi) - Some algebraic obstacles......Page 110
Algebraic Excursion: Re-expressing the powers of an algebraic number......Page 111
4.8 The transcendence of exp(Pi)......Page 113
5.1 Algebraic exponents and bases-Moving beyond e by focusing on e......Page 122
5.2 Some sketchy thoughts on the proof of Theorem 5.2......Page 124
5.3 Distilling three algebraic numbers down to one primitive element......Page 127
Algebraic Excursion: Number fields and primitive elements......Page 128
5.4 Beating the (linear) system - Constructing a polynomial via linear equations......Page 131
5.5 The proof of a real special case......Page 136
5.6 Moving out to the vast complex plane......Page 142
5.7 Widening our perspective and extending our results......Page 145
5.8 Algebraic values of algebraically independent functions......Page 149
6.1 The power of making polynomials nearly vanish......Page 156
6.2 A rational approach to irrationality......Page 158
6.3 An algebraic approach to transcendence......Page 161
6.4 Detecting subtle distinctions among the transcendent......Page 167
6.5 A critical consequence of the classification......Page 172
6.6 Which is the most popular class? Granting "Most favored number status"......Page 176
6.7 The proofs of Lemmas 6.14 and 6.15......Page 183
Algebraic Excursion: Algebraic approximations and polynomials with small moduli......Page 184
6.8 Declassified quantities: e, Pi, and the elusive T-numbers......Page 190
7.1 Transcending our beloved e^z and challenging its centrality......Page 192
7.2 A circle of ideas behind elliptic curves......Page 193
7.3 Entire periodic functions......Page 200
7.4 Pinning down p(z) and uncovering a group homomorphism......Page 203
7.5 A new transcendence result......Page 209
7.6 Exploring the gamma function and infinite products......Page 212
7.7 The proof of Theorem 7.3......Page 217
Algebraic Excursion: An introduction to the theory of complex multiplication......Page 225
8.1 Moving beyond numbers......Page 232
8.2 An intimate interlude - How to get close in function fields......Page 234
8.3 A formal search for e......Page 237
8.4 Finding the factorial in F_q[T]......Page 238
Algebraic Excursion: Irreducible polynomials over F_q......Page 241
8.5 The transcendence of e_C(l)......Page 243
8.6 A repeated look at e_C(z) through periodicity......Page 249
8.7 The transcendence of Pi_C......Page 258
8.8 Revisiting p(z) and moving beyond Carlitz......Page 259
A.1 Analytic and entire functions......Page 264
A.3 Cauchy's integral formula......Page 265
A.4 The Maximum Modulus Principle......Page 266
Acknowledgments......Page 268
Index......Page 270