Exploratory Galois Theory

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Cambridge University Press, Oct 11, 2004 - Computers - 208 pages
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Combining a concrete perspective with an exploration-based approach, Exploratory Galois Theory develops Galois theory at an entirely undergraduate level. The text grounds the presentation in the concept of algebraic numbers with complex approximations and assumes of its readers only a first course in abstract algebra. For readers with Maple or Mathematica, the text introduces tools for hands-on experimentation with finite extensions of the rational numbers, enabling a familiarity never before available to students of the subject. The text is appropriate for traditional lecture courses, for seminars, or for self-paced independent study by undergraduates and graduate students.
 

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Contents

Preliminaries
5
Algebraic Numbers Field Extensions and Minimal Polynomials
22
Working with Algebraic Numbers Field Extensions and Minimal Polynomials
39
Multiply Generated Fields
63
18 Isomorphisms from Multiply Generated Fields
78
19 Fields and Splitting Fields Generated by Arbitrarily Many Algebraic Numbers
83
20 Exercise Set 1
86
Maple and Mathematica
89
29 Exercise Set 2
149
Some Classical Topics
152
31 Cyclic Extensions over Fields with Roots of Unity
156
32 Binomial Equations
161
33 RulerandCompass Constructions
163
34 Solvability by Radicals
171
35 Characteristic p and Arbitrary Fields
177
36 Finite Fields
186

22 Exercise Set 2
100
The Galois Correspondence
103
24 The Galois Group
105
25 Invariant Polynomials Galois Resolvents and the Discriminant
115
26 Exercise Set 1
127
27 Distinguishing Numbers Determining Groups
128
Maple and Mathematica
137
Historical Note
193
Subgroups of Symmetric Groups
197
2 The Subgroups of S5
198
Bibliography
201
Index
205
Copyright

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Page 204 - Remarques relatives 1 à des classes très-étendues de quamités dom la valeur n'est ni rationnelle ni même réductible à des irrationnelles algébriques; 2 à un passage du livre des Principes où Newton calcule l'action exercée par une sphère sur un poim extérieur et Nouvelle demonstration d'un théorème sur les irrationnelles aigébriques.

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About the author (2004)

John Swallow is John T. Kimbrough Associate Professor of Mathematics at Davidson College, North Carolina. He holds a doctorate from Yale University, Connecticut for his work in Galois theory. He is the author or co-author of a dozen articles, including an essay in The American Scholar. His work has been supported by the National Science Foundation, the National Security Agency, and the Associated Colleges of the South.

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