# Algebra I (MATH 101a, Fall 2013)

 Day & Time: Tuesday and Thursday 3:30pm — 4:50am First day of class: Tuesday, September 3 Location: Goldsmith 116 Instructor: Joel Bellaiche Textbooks: For references, I will essentially use Algebra of Serge Lang (third or fourth edition -- they are almost identical). Occasionally I will refer to Basic Algebra, of Jacobson. Both books are very good, but they have a different style : Lang is quick and sometimes allusive, while Jacobson is more thorough. I will also use another book for category theory, to be precised later. Program: PART I : Groups Groups; Morphisms of groups; subgroups; generating family; cosets; Normal subgroups; Quotient groups; Simple groups; Jordan-Holder theorem; Direct and semi-direct products; extensions; Cyclic groups; finite abelian groups; Operation of a group on a set; Symmetric and alternate groups: definitions, generating familes, simplicity. Solvable groups; p-groups; Sylow subgroups and Sylow theorems; Other examples of finite simple groups; PART II : Categories Definition of a category; Examples; Fonctors; Natural transformation of functors; Equivalence of categories; Initial anf final objects of a cateory; Commutative diagrams; limits and colimits; Fibred products and coproducts; Adjoint functors; examples; PART III : Commutative rings Rings, ideals, modules; Principal rings; Factorial rings; Noetherian rings; Local rings and localization; Nakayama's lemma; Projective modules; Free modules; Modules over a principal rings Tensor product of modules; flat modules; PART IV : Fields and Galois theory Fields; Extensions of fields; finite and algebraic extensions; Normal and separable extensions; Perfect fields; Primitive element theorem; Galois theory (standard forms); Solvability of equations; Other formulations of Galois theory; Part V : Non commutative ring theory and representations of finite groups;

## Schedule of Lectures

 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)