Inverse Galois Problem

Importance: Outstanding ✭✭✭✭
Author(s): Hilbert, David
Subject: Group Theory
Recomm. for undergrads: no
Posted by: tchow
on: October 13th, 2008
Conjecture   Every finite group is the Galois group of some finite algebraic extension of $ \mathbb Q $.

This problem is one of the greatest open problems in group theory. Hilbert was the first to study it in earnest. His irreducibility theorem established a connection between Galois groups over $ \mathbb Q $ and Galois groups over $ {\mathbb Q}(x) $; the latter could be attacked by geometric methods, and in this way, Hilbert showed that the symmetric and alternating groups are Galois realizable over $ \mathbb Q $. In the 1950's, Shafarevich showed using number-theoretic methods that all finite solvable groups are Galois realizable over $ \mathbb Q $. Another spectacular result was John Thompson's realization of the Monster group as a Galois group over $ \mathbb Q $. One of Thompson's main tools was a concept he called "rigidity", a concept discovered independently by several people that continues to be important to this day. It is now known that 25 of the 26 sporadic simple groups are Galois realizable over $ \mathbb Q $ (the sole exception being the Mathieu group $ M_{23} $).


[MM] Gunter Malle and B. Heinrich Matzat, Inverse Galois Theory, Springer, 1999.

[V] Helmut Völklein, Groups as Galois Groups: An introduction, Cambridge University Press, 1996.

* indicates original appearance(s) of problem.