Algebraic Equations

By: EDGAR DEHN

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Paperback | 1 August 2017

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This text focuses on the basics of algebraic theory, giving detailed explanations of integral functions, permutations, and groups, and Lagrange and Galois theory. Many numerical examples with complete solutions. 1930 edition.

Meticulous and complete, this presentation of Galois' theory of algebraic equations is geared toward upper-level undergraduate and graduate students. The theories of both Lagrange and Galois are developed in logical rather than historical form and given a thorough exposition. For this reason, Algebraic Equations is an excellent supplementary text, offering students a concrete introduction to the abstract principles of Galois theory. Of further value are the many numerical examples throughout the book, which appear with complete solutions.

A third of the text focuses on the basic ideas of algebraic theory, giving detailed explanations of integral functions, permutations, and group in addition to a very clear exposition of the symmetric group and its functions. A study of the theory of Lagrange follows. After a discussion of various groups (including isomorphic, transitive, and Abelian groups), a detailed study of Galois theory covers the properties of the Galoisian function, the general equation, reductions of the group, natural irrationality, and other features. The book concludes with the application of Galoisian theory to the solution of such special equations as Abelian, cyclic, metacyclic, and quintic equations.

About the Author

Edgar Dehn was an Instructor in Mathematics at Columbia University.

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List of Theorems	p. viii
List of Adopted Conventions	p. ix
Integral Function
Interpolation	p. 1
Division	p. 3
Reduction	p. 6
Primitive function	p. 7
Linear factors	p. 8
Equations and Permutations
Discovery of Lagrange	p. 11
Solution of cubic	p. 13
Connection with permutations	p. 15
Algebra of Permutations
Notation	p. 17
Degree	p. 18
Combination	p. 19
Order	p. 21
Association	p. 23
Inverse	p. 24
Identity	p. 26
Group and Subgroup
Group	p. 27
Subgroup	p. 29
Conjugate subgroups	p. 32
Rule of transforms	p. 34
Normal subgroup	p. 36
Symmetric Group and Its Functions
Generator	p. 38
Symmetric sum	p. 40
Computation of symmetric sum	p. 41
Another computation	p. 44
Resultant	p. 46
Resultant as determinant	p. 47
Discriminant	p. 50
Composition of Symmetric Group
Composition-series	p. 53
Alternating function	p. 54
Alternating group	p. 56
Composition of S and A	p. 59
Subgroups of S and A	p. 61
Group on functions	p. 64
Theory of Lagrange
Resolvent equation	p. 66
Lagrange's Theorem	p. 67
Lagrange's Theorem, Continued	p. 70
Plan of Lagrange	p. 74
Lagrange's solvent	p. 75
Special case of solvents	p. 78
Limits of Lagrange's plan	p. 80
General Equations
Quadratic equation	p. 82
Cubic equation	p. 83
Cubic equation, Continued	p. 85
Cubic equation, Continued	p. 87
Cubic equation, Continued	p. 88
Biquadratic equation	p. 90
Biquadratic equation, Continued	p. 93
Biquadratic equation, Continued	p. 95
More About Groups
Isomorphic groups	p. 99
Transitive group	p. 101
Imprimitive group	p. 104
Quotient-group	p. 107
Subgroups of quotient-group	p. 110
Maximum normal subgroup	p. 111
Constancy of composition-factors	p. 113
Abelian group	p. 115
Theorem of Cauchy	p. 118
Metacyclic group	p. 120
Note on abstract group	p. 126
Domain
Algebraic domain	p. 129
Algebraic domain, Continued	p. 131
Conjugate domains	p. 133
Conjugate domains, Continued	p. 135
Normal domain	p. 136
Theory of Galois
Special equation	p. 139
Galoisian function	p. 140
Galoisian resolvent	p. 143
Galoisian group	p. 145
Properties of Galoisian group	p. 146
Plan of Galois	p. 149
General equation	p. 153
Duality of plans	p. 155
Irreducible equation	p. 156
Applications	p. 158
Imprimitive equation	p. 161
Reduction of group	p. 163
Natural irrationality	p. 169
Special Equations
Abelian equation	p. 171
Cyclic equation	p. 172
Roots of unity	p. 175
Congruence	p. 176
Fermat's Theorem	p. 179
Cyclotomic equation	p. 182
Discriminant of cyclotomic equation	p. 184
Applications	p. 185
Metacyclic equation	p. 190
Quintic equation	p. 195
Index	p. 201
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Algebraic Equations

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