Background

The inverse Galois problem asks whether every finite group can be realized as the Galois group of a field extension of the rational numbers (ℚ). This is a major unsolved problem in algebra that has attracted significant attention from mathematicians. While many finite groups are known to be realizable as Galois groups over ℚ, including all finite solvable groups, all simple sporadic groups (except possibly the Mathieu group M₂₃), and all symmetric and alternating groups, the general case remains unproven.

Resolution Criteria

This market will resolve YES if a mathematical proof is published and accepted by the mathematical community demonstrating that every finite group can be realized as a Galois group of a field extension of ℚ. The market will resolve NO if a counterexample is found - that is, if a finite group is proven to not be realizable as a Galois group over ℚ.

Considerations

• This is a long-standing open problem in mathematics that has resisted solution for over a century

• A resolution would represent a significant breakthrough in Galois theory and algebra

• The problem has been partially solved for many important classes of finite groups

• Given the complexity and fundamental nature of this problem, any claimed solution would likely undergo extensive peer review before being widely accepted by the mathematical community