## GALOIS THEORY

Teachers

##### Mario MainardisFernando JosÉ Barrera Esteban
2
6
First Period
###### Course Type
Characterizing
Prerequisites.  Algebra and Geometry 1
Teaching Methods. Lessons and 12 supplementary problem sessions all held by the teacher
Verification of Learning. Written test at the end of the course.  The test consists of a short exposition (about 2000 words) and the solutions of two or three exercises. In the exposition, on one of the arguments exposed during the course, the student will have to show to have understood the basic concepts and arguments and be able to expose them correctly. In the exercises the student will have to show that he/she is able to apply the techniques exposed during the course. The student will have approximatley to hours to complete the written test
More Information. Moodle web page available  On request the lectures can be held in English or German
##### Objectives
The student should be confident with the basic notions of group actions and be able to apply them, in particular, to field extensions.
##### Contents
The course consists in a classical exposition of Galois Theory. In the first part we’ll develop basic concepts and methods of group theory and rings, in particular group actions, solvability and the classification of finitely generated modules on a principal ideal domain. In the second part we’ll apply this machinery to galois duality to prove Galois criterion of solvability.  Detailed programme: Ruler and compass constructions. Group actions, G-sets, First Homomorphism Theorem for G-sets, orbit equation, nilpotency of finite p-groups, Sylow’s theorems, solvable groups, Fitting’s theorem, finitely generated modules over a principal ideal domain, fundamentals of Galois theory, Galois extensions, separability, and pure inseparability, finite fields, Galois criterion for solvability by radicals.
##### Texts
1) Teacher’s lecture notes available at https://users.dimi.uniud.it/~mario.mainardis/LIBRO02.pdf 2) Title: Fields and Rings, Author: Irving Kaplansky, Publisher: Chicago University Press. 3) Title: Basic Algebra 1, Author: Nathan Jacobson, Publisher W. H. Freeman & Co.