Elements of Algebra: Geometry, Numbers, Equations

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Springer Science & Business Media, Jul 20, 2001 - Mathematics - 184 pages
Algebra 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.
 

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Contents

I
3
IV
7
V
9
VI
12
VII
13
VIII
15
IX
16
X
17
XLIX
88
L
91
LI
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LII
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LIII
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LIV
100
LV
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LVI
103

XI
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XIII
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XIV
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XV
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XVI
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XVII
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XVIII
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XIX
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XXI
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XXIII
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XXVII
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XXVIII
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XXX
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XXXI
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XXXIII
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XXXIV
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XXXV
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XXXVI
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XXXVII
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XXXVIII
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XXXIX
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XL
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XLI
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XLII
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XLIV
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XLV
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XLVI
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XLVII
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XLVIII
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LVII
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LVIII
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LIX
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LX
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LXI
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LXII
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LXIII
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LXIV
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LXV
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LXVI
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LXVII
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LXVIII
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LXIX
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LXXI
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LXXII
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LXXIII
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LXXIV
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LXXV
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LXXVI
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LXXVII
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LXXVIII
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LXXIX
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LXXX
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LXXXI
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LXXXII
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LXXXIII
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LXXXIV
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LXXXV
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LXXXVI
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LXXXVII
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LXXXVIII
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LXXXIX
164
XC
172
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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.

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