PART I : Groups
Part V : Non commutative ring theory and representations of finite groups; |

Class |
Topic |
Readings |
Exercises |

Sept 3 | Presentation of the course. Groups: definitions; subgroups; subgroup generated by a subset; morphisms; | ||

Sept 10 | Groups: cosets; normal subgroups; | ||

Sept 12 | Groups: factor groups; canonical isomorphisms; Tower of groups; Schreier's theorem | Exercises | |

Sept 17 | Groups: Jordan-Holder theorem; extensions; direct and semi-direct product; | ||

Sept 24 | Semi-direct product and split extensions | ||

Oct 1 | Groups: cyclic and dihedral groups; order of an element; | ||

Oct 3 | Torsion-free and free abelian groups | ||

Oct 8 | Finitely generated abelian groups. | Exercises Solutions | |

Oct 10 | Groups: Operations of a group on a set; | ||

Oct 15 | Symmetric groups; alternating groups; | ||

Oct 17 | Alternating groups; | ||

Oct 22 | Simplicity of the alternating group. finite groups (Lagrange's theorem); | Exercises | |

Oct 24 | Sylow's theorems and applications. | ||

Oct 29 | Categories. Diagrams. Initial and final objects. | ||

Oct 31 | Categories: limits of diagrams | Exercises | |

Nov 5 | Categories: monomorphisms and epimorphisms. Functors. Natural Transformations | ||

Nov 7 | Rings, domains, fields: definitions | ||

Nov 12 | Rings: ideals; principal ideal domain; noetherian rings; | ||

Nov 14 | Rings: prime and maximal ideal; irreducible elements; UFD's; | ExercisesSolutions | |

Nov 19 | Rings: rings of polynomials; Hilbert's basis theorem; | ||

Nov 21 | Rings: fraction field of a domain; field of rational functions; | ||

Nov 25 (Monday) | Fields: extensions of field; degree of a finite extension; multiplicativity of degree; | ||

Nov 26 | Fields: algebraic and transcendental extensions; | Exercises | |

Dec 3 | Fields: normal and separable extensions; | ||

Dec 5 | Fields: Galois extensions and Galois groups; Galois theory; | Exercises | |

Supplementary session (Dec 10, 3pm) | Fields: Applications of Galois theory to the solution of algebraic equations and to geometric constructions. | Exercises(not due) | |