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).

Lecturer
Kiselev Denis Dmitrievich

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