A Field Guide to Algebra

✍ Scribed by Antoine Chambert-Loir

• Publisher: Springer Science & Business Media
• Year: 2007
• Tongue: English
• Leaves: 199
• Category: Library

✦ Synopsis

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This is a small book on algebra where the stress is laid on the structure of ?elds, hence its title. Youwillhearaboutequations,bothpolynomialanddi?erential,andabout the algebraic structure of their solutions. For example, it has been known for centuries how to explicitely solve polynomial equations of degree 2 (Baby- nians, many centuries ago), 3 (Scipione del Ferro, Tartaglia, Cardan, around th 1500a.d.), and even 4 (Cardan, Ferrari,xvi century), using only algebraic operations and radicals (nth roots). However, the case of degree 5 remained unsolved until Abel showed in 1826 that a general equation of degree 5 cannot be solved that way. Soon after that, Galois de?ned the group of a polynomial equation as the group of permutations of its roots (say, complex roots) that preserve all algebraicidentitieswithrationalcoe?cientssatis?edbytheseroots.Examples of such identities are given by the elementary symmetric polynomials, for it is well known that the coe?cients of a polynomial are (up to sign) elementary symmetric polynomials in the roots. In general, all relations are obtained by combining these, but sometimes there are new ones and the group of the equation is smaller than the whole permutation group. Galois understood how this symmetry group can be used to characterize the solvability of the equation. He de?ned the notion of solvable group and showed that if the group of the equation is solvable, then one can express its roots with radicals, and conversely.

✦ Table of Contents

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PIÑERA CHUALO
Contents
Preface
1 Field extensions
1.1 Constructions with ruler and compass
1.2 Fields
1.3 Field extensions
1.4 Some classical impossibilities
1.5 Symmetric functions
1.6 Appendix: Transcendence of e and π
Exercises
2 Roots
2.1 Ring of remainders
2.2 Splitting extensions
2.3 Algebraically closed fields; algebraic closure
2.4 Appendix: Structure of polynomial rings
2.6 Appendix: Puiseux’s theorem
Exercises
3 Galois theory
3.1 Homomorphisms of an extension in an algebraic closure
3.2 Automorphism group of an extension
3.3 The Galois group as a permutation group
3.4 Discriminant; resolvent polynomials
3.5 Finite fields
Exercises
4 A bit of group theory
4.1 Groups (quick review of basic definitions)
4.2 Subgroups
4.3 Group actions
4.4 Normal subgroups; quotient groups
4.5 Solvable groups; nilpotent groups
4.6 Symmetric and alternating groups
4.7 Matrix groups
Exercises
5 Applications
5.1 Constructibility with ruler and compass
5.2 Cyclotomy
5.3 Composite extensions
5.4 Cyclic extensions
5.5 Equations with degrees up to 4
5.6 Solving equations by radicals
5.7 How (not) to compute Galois groups
5.8 Specializing Galois groups
5.9 Hilbert’s irreducibility theorem
Exercises
6 Algebraic theory of differential equations
6.1 Differential fields
6.2 Differential extensions; construction of derivations
6.3 Differential equations
6.4 Picard-Vessiot extensions
6.5 The differential Galois group; examples
6.6 The differential Galois correspondence
6.7 Integration in finite terms, elementary extensions
6.8 Appendix: Hilbert’s Nullstellensatz
Exercises
Examination problems
References
Index