| Antoine Chambert-Loir taught this book when he was Professor at École Polytechnique, Palaiseau, France. He is now Professor at Université de Rennes 1. |
| 1 Field extensions Constructions with ruler and compass Fields Field extensions Some classical impossibilities Symmetric functions Appendix: Transcendence of e and π Exercises 2 Roots Ring of remainders Splitting extensions Algebraically closed fields; algebraic closure Appendix: Structure of polynomial rings Appendix: Quotient rings Appendix: Puiseux's theorem Exercises 3 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 Exercises 4 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 Exercises 5 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 Exercises 6 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 Exercises Examination problems References Index |
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