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Galois Theory : Graduate Texts in Mathematics - Harold M. Edwards

Galois Theory

Graduate Texts in Mathematics

Hardcover Published: May 1984
ISBN: 9780387909806
Number Of Pages: 154

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This book is an introduction to Galois theory along the lines of Galois' "Memoir on the Conditions for Solvability of Equations by Radicals." Some antecedents of Galois theory in the works of Gauss, Lagrange, Vandemonde, Newton, and even the ancient Babylonians, are explained in order to put Galois' main ideas in their historical setting. The modern formulation of the theory is also explained. The book contains many exercises - with answers - and an English translation of Galois' memoir.

Acknowledgments
Galois
Influence of Lagrange
Quadratic equations
1700 B.C. to A.D. 1500
Solution of cubic
Solution of quartic
Impossibility of quintic
Newton
Symmetric polynomials in roots
Fundamental theorem on symmetric polynomials
Proof
Newton's theorem
Discriminants
Solution of cubic
Lagrange and Vandermonde
Lagrange resolvents
Solution of quartic again
Attempt at quintic
Lagrange's Reflexions
Cyclotomic equations
The cases n = 3, 5
n = 7, 11
General case
Two lemmas
Gauss's method
p-gons by ruler and compass
Summary
Resolvents
Lagrange's theorem
Proof
Galois resolvents
Existence of Galois resolvents
Representation of the splitting field as K(t)
Simple algebraic extensions
Euclidean algorithm
Construction of simple algebraic extensions
Galois' method
Review
Finite permutation groups
Subgroups, normal subgroups
The Galois group of an equation
Examples
Solvability by radicals
Reduction of the Galois group by a cyclic extension
Solvable groups
Reduction to a normal subgroup of index p
Theorem on solution by radicals (assuming roots of unity)
Summary
Splitting fields
Fundamental theorem of algebra (so-called)
Construction of a splitting field
Need for a factorization method
Three theorems on factorization methods
Uniqueness of factorization of polynomials
Factorization over Z
Over Q
Gauss's lemma, factorization over Q
Over transcendental extensions
Of polynomials in two variables
Over algebraic extensions
Final remarks
Review of Galois theory
Fundamental theorem of Galois theory (so-called)
Galois group of x[superscript p] - 1 = 0 over Q
Solvability of the cyclotomic equation
Theorem on solution by radicals
Equations with literal coefficients
Equations of prime degree
Galois group of x[superscript n] - 1 = 0 over Q
Proof of the main proposition
Deduction of Lemma 2 of 24
Memoir on the Conditions for Solvability of Equations by Radicals, by Evariste Galois
Synopsis
Groups
Answers to Exercises
List of Exercises
References
Index
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387909806
ISBN-10: 038790980X
Series: Graduate Texts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 154
Published: May 1984
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.4 x 15.6  x 1.91
Weight (kg): 0.4

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