Dover Books on Mathematics

By: M. M. Postnikov, Ann Swinfen (Translator)

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Paperback | 2 February 2004

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109 Pages

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Written by a prominent mathematician, this text offers advanced undergraduate and graduate students a virtually self-contained treatment of the basics of Galois theory. The source of modern abstract algebra and one of abstract algebra's most concrete applications, Galois theory serves as an excellent introduction to group theory and provides a strong, historically relevant motivation for the introduction of the basics of abstract algebra.
This two-part treatment begins with the elements of Galois theory, focusing on related concepts from field theory, including the structure of important types of extensions and the field of algebraic numbers. A consideration of relevant facts from group theory leads to a survey of Galois theory, with discussions of normal extensions, the order and correspondence of the Galois group, and Galois groups of a normal subfield and of two fields. The second part explores the solution of equations by radicals, returning to the general theory of groups for relevant facts, examining equations solvable by radicals and their construction, and concluding with the unsolvability by radicals of the general equation of degree n ≥ 5.

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Non-Fiction Mathematics Algebra Arts & Entertainment Architecture

Foreword	p. vii
Preface	p. viii
The Elements of Galois Theory
The Elements of Field Theory	p. 1
Preliminary remarks	p. 1
Some important types of extensions	p. 2
The minimal polynomial. The structure of simple algebraic extensions	p. 4
The algebraic nature of finite extensions	p. 6
The structure of composite algebraic extensions	p. 7
Composite finite extensions	p. 9
The theorem that a composite algebraic extension is simple	p. 12
The field of algebraic numbers	p. 14
The composition of fields	p. 14
Necessary Facts from the Theory of Groups	p. 16
The definition of a group	p. 16
Subgroups, normal divisors and factor groups	p. 18
Homomorphic mappings	p. 21
Galois Theory	p. 25
Normal extensions	p. 25
Automorphisms of fields. The Galois group	p. 28
The order of the Galois group	p. 31
The Galois correspondence	p. 35
A theorem about conjugate elements	p. 38
The Galois group of a normal subfield	p. 39
The Galois group of the composition of two fields	p. 41
The Solution of Equations by Radicals
Additional Facts from the General Theory of Groups	p. 45
A generalization of the homomorphism theorem	p. 45
Normal series	p. 46
Cyclic groups	p. 49
Solvable and Abelian groups	p. 52
Equations Solvable by Radicals	p. 58
Simple radical extensions	p. 58
Cyclic extensions	p. 60
Radical extensions	p. 65
Normal fields with solvable Galois group	p. 68
Equations solvable by radicals	p. 71
The Construction of Equations Solvable by Radicals	p. 73
The Galois group of an equation as a group of permutations	p. 73
The factorization of permutations into the product of cycles	p. 75
Even permutations. The alternating group	p. 79
The structure of the alternating and symmetric groups	p. 81
An example of an equation with Galois group the symmetric group	p. 85
A discussion of the results obtained	p. 89
The Unsolvability by Radicals of the General Equation of Degree n [greater than or equal] 5	p. 92
The field of formal power series	p. 92
The field of fractional power series	p. 97
The Galois group of the general equation of degree n	p. 101
The solution of equations of low degree	p. 105
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Dover Books on Mathematics

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