inverse Galois problem

The inverse Galois problem is extremely to , yet one of the hardest problems for theorists and . It generally asks for descriptions of the of groups that can occur as Galois groupsMathworldPlanetmath. Of course, a significantly more precise formulation is required, for example, because of a result that that every Galois group is profinite, and every profinite group is a Galois group. Also ambiguous is what field(s) we allow ourselves to include when computing the Galois group. Unfortunately, many of these related questions all go under the heading “the inverse Galois problem,” so care must be taken to determine an exact formulation of the question being asked.

As an example of a partial solution to this question, it is known that every finite abelian group occurs as the Galois group of an extensionPlanetmathPlanetmath over (by the Kronecker-Weber theoremMathworldPlanetmath), though it is not known whether or not this is true for every finite (not necessarily abelianMathworldPlanetmath) group. This latter question can also be phrased in of the absolute Galois group: “Does every finite groupMathworldPlanetmath occur as a quotient groupMathworldPlanetmath of the absolute Galois group Gal(¯/)”? Thus, an answer to this question would not only reveal information about the nature of finite Galois groups, but also shed light on one of the most elusive objects in all of algebra and number theoryMathworldPlanetmath.

It is also known (see Shafarevich’ theorem) that every solvable groupMathworldPlanetmath occurs as a Galois group.

Title inverse Galois problem
Canonical name InverseGaloisProblem
Date of creation 2013-03-22 15:01:14
Last modified on 2013-03-22 15:01:14
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 34M50
Classification msc 13B05
Classification msc 11R32
Related topic ShafarevichsTheorem