Galois Cohomology, An Undergraduate Course

Timetable

Module aim (aims)

Galois Cohomology is a one of the most important tools in Arithmetic Algebraic Geometry. The purpose of this course is a presentation of basic methods and results of Galois Cohomology. This course will also discuss results from other areas of Mathematics that are in the foundations of Galois Cohomology.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Basic courses in Linear Algebra and  Abstract Algebra.

Syllabus

Week 1: Field Theory     

Week 2: Field theory     

Week 3: Galois Theory  

Week 4: Galois Theory  

Week 5: Modules, complexes, cohomology.       

Week 6: The group ring. Cohomology of groups.              

Week 7: Cohomology of groups.              Shapiro's lemma.

Week 8: Inflation, restriction and corestriction.

Week 9: Topological groups. Inverse systems.  

Week 10: Profinite groups. Discrete G-modules.             

Week 11: Galois Cohomology; basic properties

Week 12: Galois Cohomology   

Week 13: Applications of Galois Cohomology

Week 14: Applications of Galois Cohomology                    

Week 15: Galois cohomology and Arithmetic Algebraic Geometr

Reading list

1. J.W.S. Cassels, A. Frochlich „Algebraic Number Theory” , Academic Press, (1967)
2. J-P Serre, „Galois Cohomology”, Springer 1997
3. S. Shatz, „Profinite Groups, Arithmetic and Geometry”, Annals of Math. Studies, Princeton Univ. Press 1972
4. L. Washington, „Galois Cohomology”, article in the book „Modular forms and Fermat's last theorem”, Springer 1997