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July 15, 2025, 10:44 a.m.
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SOURCE: Chambert-Loir A. A Field Guide to Algebra 2005
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COVER
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MEDIAINFO
Textbook in PDF format This is a small book on algebra where the stress is laid on the structure of fields, hence its title. You will hear about equations, both polynomial and differential, and about the algebraic structure of their solutions. Field extensions. Constructions with ruler and compass. Fields. Field extensions. Some classical impossibilities. Symmetric functions. Appendix: Transcendence of e and π. Roots. Ring of remainders. Splitting extensions. Algebraically closed fields; algebraic closure. Appendix: Structure of polynomial rings. Appendix: Quotient rings. Appendix: Puiseux’s theorem. Galois theory. Homomorphisms of an extension in an algebraic closure. Automorphism group of an extension. The Galois group as a permutation group. Discriminant; resolvent polynomials. Finite fields. A bit of group theory. Groups (quick review of basic definitions). Subgroups. Group actions. Normal subgroups; quotient groups. Solvable groups; nilpotent groups. Symmetric and alternating groups. Matrix groups. Applications. Constructibility with ruler and compass. Cyclotomy. Composite extensions. Cyclic extensions. Equations with degrees up to 4. Solving equations by radicals. How (not) to compute Galois groups. Specializing Galois groups. Hilbert’s irreducibility theorem. Algebraic theory of differential equations. Differential fields. Differential extensions; construction of derivations. Differential equations. Picard-Vessiot extensions. The differential Galois group; examples. The differential Galois correspondence. Integration in finite terms, elementary extensions. Appendix: Hilbert’s Nullstellensatz
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