Inverse Galois Problem

Importance: Outstanding ✭✭✭✭
Author(s): Hilbert, David
Subject: Group Theory
Recomm. for undergrads: no
Posted by: tchow
on: October 13th, 2008

\begin{conjecture} Every finite group is the Galois group of some finite algebraic extension of $\mathbb Q$. \end{conjecture}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{} 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 $M_{23}$).


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries) [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.