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Course by D. D. Kiselev "Inverse Galois Theory: embedding problem"
September 8–December 8, 2020, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)

Inverse Galois Theory is a beautiful and rather complicated part of modern algebra. It deals with sufficient conditions for a given finite group $G$ to be realised as Galois group of some Galois field extension $K/k$ over fixed field $k$. If $k$ is a number field (a finite extension of rational number field $\mathbb{Q}$), then the solution of such problem is still to be completed. At the same time, for the case of $1$-dimensional local field $k$ (a finite extension of $p$-adic field $\mathbb{Q}_p$) the solution exists (for the case $p > 2$ and for the case $p = 2$ with the additional condition $\sqrt{-1} \in k$).

The goal of the course is to give for students an introduction to inverse Galois Theory via embedding problem and also to give a positive solution of Inverse Galois problem for all finite groups of odd order over number fields (in particular, over rational number field $\mathbb{Q}$). The above result is a partial case of I. R. Shafarevich's theorem, which is devoted to realizability of any finite solvable group as Galois group over arbitrary number field $k$, but we are going to obtain this result by using less complicated proof for such case.

It is supposed that students will familiarize themselves with preliminaries of algebraic number theory, group cogomology, Brauer group, Galois Theory in the context of Galois correspondence. The required results for the course will be revised if necessary but without proof.

The course is expected to contain 12-14 lectures.

Financial support. The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2019-1614).

Program

RSS: Forthcoming seminars

Lecturer
Kiselev Denis Dmitrievich

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center


Course by D. D. Kiselev "Inverse Galois Theory: embedding problem", September 8–December 8, 2020

December 8, 2020 (Tue)
1. Lecture 14. Inverse Galois Theory: embedding problem
D. D. Kiselev
December 8, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

December 1, 2020 (Tue)
2. Lecture 13. Inverse Galois Theory: embedding problem
D. D. Kiselev
December 1, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

November 24, 2020 (Tue)
3. Lecture 12. Inverse Galois Theory: embedding problem
D. D. Kiselev
November 24, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

November 17, 2020 (Tue)
4. Lecture 11. Inverse Galois Theory: embedding problem
D. D. Kiselev
November 17, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

November 10, 2020 (Tue)
5. Lecture 10. Inverse Galois Theory: embedding problem
D. D. Kiselev
November 10, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

November 3, 2020 (Tue)
6. Lecture 9. Inverse Galois Theory: embedding problem
D. D. Kiselev
November 3, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

October 27, 2020 (Tue)
7. Lecture 8. Inverse Galois Theory: embedding problem
D. D. Kiselev
October 27, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

October 20, 2020 (Tue)
8. Lecture 7. Inverse Galois Theory: embedding problem
D. D. Kiselev
October 20, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

October 13, 2020 (Tue)
9. Lecture 6. Inverse Galois Theory: embedding problem
D. D. Kiselev
October 13, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

October 6, 2020 (Tue)
10. Lecture 5. Inverse Galois Theory: embedding problem
D. D. Kiselev
October 6, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

September 29, 2020 (Tue)
11. Lecture 4. Inverse Galois Theory: embedding problem
D. D. Kiselev
September 29, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

September 22, 2020 (Tue)
12. Lecture 3. Inverse Galois Theory: embedding problem
D. D. Kiselev
September 22, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

September 15, 2020 (Tue)
13. Lecture 2. Inverse Galois Theory: embedding problem
D. D. Kiselev
September 15, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  

September 8, 2020 (Tue)
14. Lecture 1. Inverse Galois Theory: embedding problem
D. D. Kiselev
September 8, 2020 18:00, Steklov Mathematical Institute, Conference hall, 9th floor (8 Gubkina)
D. D. Kiselev
  
 
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