Elements of Algebra: Geometry, Numbers, EquationsAlgebra is abstract mathematics - let us make no bones about it - yet it is also applied mathematics in its best and purest form. It is not abstraction for its own sake, but abstraction for the sake of efficiency, power and insight. Algebra emerged from the struggle to solve concrete, physical problems in geometry, and succeeded after 2000 years of failure by other forms of mathematics. It did this by exposing the mathematical structure of geometry, and by providing the tools to analyse it. This is typical of the way algebra is applied; it is the best and purest form of application because it reveals the simplest and most universal mathematical structures. The present book aims to foster a proper appreciation of algebra by showing abstraction at work on concrete problems, the classical problems of construction by straightedge and compass. These problems originated in the time of Euclid, when geometry and number theory were paramount, and were not solved until th the 19 century, with the advent of abstract algebra. As we now know, alge bra brings about a unification of geometry, number theory and indeed most branches of mathematics. This is not really surprising when one has a historical understanding of the subject, which I also hope to impart. |
Contents
I | 3 |
III | 7 |
IV | 9 |
V | 12 |
VI | 13 |
VII | 15 |
VIII | 16 |
IX | 17 |
XLVIII | 88 |
XLIX | 91 |
LI | 93 |
LII | 96 |
LIII | 99 |
LIV | 100 |
LV | 102 |
LVI | 103 |
X | 20 |
XI | 22 |
XII | 24 |
XIII | 25 |
XIV | 27 |
XV | 28 |
XVI | 30 |
XVII | 31 |
XVIII | 34 |
XIX | 36 |
XX | 40 |
XXII | 42 |
XXIII | 43 |
XXIV | 44 |
XXV | 46 |
XXVI | 48 |
XXVII | 50 |
XXVIII | 53 |
XXIX | 55 |
XXX | 59 |
XXXII | 60 |
XXXIII | 63 |
XXXIV | 64 |
XXXV | 65 |
XXXVI | 67 |
XXXVII | 69 |
XXXVIII | 71 |
XXXIX | 73 |
XL | 75 |
XLI | 78 |
XLIII | 79 |
XLIV | 80 |
XLV | 83 |
XLVI | 85 |
XLVII | 87 |
LVII | 105 |
LVIII | 107 |
LIX | 110 |
LX | 111 |
LXI | 113 |
LXII | 114 |
LXIII | 117 |
LXIV | 118 |
LXV | 121 |
LXVI | 123 |
LXVII | 124 |
LXVIII | 127 |
LXIX | 130 |
LXXI | 132 |
LXXII | 135 |
LXXIII | 137 |
LXXIV | 138 |
LXXV | 140 |
LXXVI | 141 |
LXXVII | 143 |
LXXVIII | 145 |
LXXIX | 148 |
LXXX | 150 |
LXXXI | 152 |
LXXXII | 154 |
LXXXIII | 156 |
LXXXIV | 157 |
LXXXV | 158 |
LXXXVI | 159 |
LXXXVII | 161 |
162 | |
164 | |
172 | |
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Common terms and phrases
a₁ abelian group algebraic numbers b₁ called Chinese remainder theorem complex numbers concept congruence classes conjugate Corollary correspondence cosets cyclotomic Dedekind Deduce from Exercise defined definition degree divides division Eisenstein's criterion equation Euclidean algorithm example fact factor Fermat prime Fermat's little theorem finite Fix(H fixed follows fundamental theorem Gal(E Galois group Galois theory Gauss gcd(m generalisation geometry group G hence homomorphism induction inverse irrational irreducible over Q irreducible polynomial isomorphism lemma Mathematics multiplication n-gon natural numbers nontrivial nonzero normal over F normal subgroup nth root number theory one-to-one p-gon permutations prime divisor prime factorisation primitive element problem proof prove pth roots Q(ao Q(Sn quadratic quotient radical extension rational numbers real numbers regular n-gon relatively prime root field root of unity rotation satisfied solvable by radicals square roots symmetry trisection Z/nZ Z/pZ
Popular passages
Page 162 - Demonstration de l'impossibilite' de la resolution alge'brique des e'quations ge'ne'rales qui passent le quatrieme degre'
Page 164 - Euler, L. (1750). Letter to Goldbach, 9 June 1750. In Fuss (1968), I, 521-524. Euler, L. (1752). Elementa doctrinae solidorum. Novi Comm. Acad. Sci. Petrop., 4, 109-140. In his Opera Omnia, ser. 1, 26: 71-93. Euler, L.
References to this book
Oxford Users' Guide to Mathematics Eberhard Zeidler,W. Hackbusch,Hans Rudolf Schwarz No preview available - 2004 |
Proofs and Fundamentals: A First Course in Abstract Mathematics Ethan D. Bloch No preview available - 2000 |