Elements of Algebra: Geometry, Numbers, Equations
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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algebraic angle automorphisms called Chapter closed coefficients complex compute concept congruence classes conjugate consider constructible contains continuous Corollary correspondence cosets course cyclic Dedekind Deduce defined definition divides division divisor edition elements equal equation equivalent example Exercises existence express extension fact factor factorisation Fermat field Figure finite fixed follows function fundamental Gal(E Galois group Gauss geometry given gives hence homomorphism identity important induction integers inverse irreducible isomorphism least lemma linear Mathematics means multiplication n-gon namely natural normal subgroup obtained one-to-one particular permutations polynomial possible preserves prime primitive problem proof properties prove quadratic radical extension rational regular remainder result ring root of unity roots rotation satisfied Second Section side solution solvable solve square Suppose symmetry theorem theory unique
Page 162 - Demonstration de l'impossibilite' de la resolution alge'brique des e'quations ge'ne'rales qui passent le quatrieme degre'
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