12.1 Inverse Galois problem

The inverse Galois problem asks if every finite group is the Galois group of a field extension over the rationals. It's a key open question in modern Galois theory, connecting group theory, field theory, and algebraic geometry.

Solving this problem would fully characterize which finite groups can be Galois groups over the rationals. While progress has been made for specific group types, a complete solution remains elusive, making it an active area of research in mathematics.

The Inverse Galois Problem

Definition and Significance

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• The inverse Galois problem asks whether every finite group appears as the Galois group of some Galois extension of the rational numbers
• Named after Évariste Galois, who laid the foundations of Galois theory in the early 19th century
• Solving the inverse Galois problem would provide a complete characterization of the finite groups that can arise as Galois groups over the rationals
• One of the central open problems in modern Galois theory
  • Has connections to various areas of mathematics (number theory, algebraic geometry, representation theory)
• A complete solution would have significant implications for understanding the structure and properties of finite groups and their realizations as Galois groups

Open Problem in Modern Mathematics

• The inverse Galois problem remains unsolved despite significant progress in specific cases
  • Affirmative solutions for abelian groups, symmetric groups, alternating groups, and many simple groups
  • Open for many classes of finite groups, particularly non-solvable groups and groups with complex structure
• Connections to other areas of mathematics make the inverse Galois problem a central question in modern Galois theory
  • Number theory: relates to the study of algebraic number fields and their Galois groups
  • Algebraic geometry: techniques from algebraic geometry (Belyi's theorem, Riemann surfaces) are used to construct Galois extensions
  • Representation theory: regular inverse Galois problem involves the study of group representations and their realizations over the rationals

Inverse Galois Problem and Field Extensions

Equivalent Formulation

• The inverse Galois problem is equivalent to determining whether every finite group can be realized as the Galois group of a Galois extension of the rational numbers
• Constructing a Galois extension with a prescribed Galois group involves finding a polynomial over the rationals whose splitting field has the desired group as its Galois group
  • Example: constructing a Galois extension with Galois group isomorphic to the symmetric group $S_n$ requires finding an irreducible polynomial of degree $n$ with prescribed properties

Tools and Techniques

• Hilbert's irreducibility theorem provides a tool for constructing Galois extensions with prescribed Galois groups
  • Shows that irreducible polynomials with certain properties exist over the rationals
  • Used to construct Galois extensions with Galois groups isomorphic to symmetric groups and alternating groups
• The Shafarevich conjecture, which states that every finite solvable group is the Galois group of some Galois extension of the rationals, is a partial solution to the inverse Galois problem for solvable groups
  • Proved by Shafarevich using techniques from algebraic number theory and class field theory
• Constructing Galois extensions with prescribed non-solvable Galois groups (Monster group, Mathieu groups) remains a challenging open problem in inverse Galois theory
  • Requires advanced techniques from algebraic geometry, representation theory, and group theory

Progress Towards Solving the Inverse Galois Problem

Affirmative Results for Specific Classes of Groups

• The inverse Galois problem has been solved affirmatively for various classes of finite groups
  • Abelian groups: every finite abelian group is the Galois group of a Galois extension of the rationals (Kronecker-Weber theorem)
  • Symmetric groups: every symmetric group $S_n$ is the Galois group of a Galois extension of the rationals (Hilbert's irreducibility theorem)
  • Alternating groups: every alternating group $A_n$ is the Galois group of a Galois extension of the rationals (Hilbert's irreducibility theorem)
  • Many simple groups: the rigidity method has been used to construct Galois extensions with Galois groups isomorphic to various simple groups (sporadic simple groups, finite simple groups of Lie type)
• Shafarevich's theorem proves that every finite solvable group is the Galois group of some Galois extension of the rationals
  • Provides a complete solution to the inverse Galois problem for solvable groups
  • Uses techniques from algebraic number theory and class field theory

Regular Inverse Galois Problem

• The regular inverse Galois problem asks whether every finite group appears as the Galois group of a regular extension of the rationals
  • A regular extension is a Galois extension where the Galois group acts freely on the roots of a generating polynomial
• The regular inverse Galois problem has been solved affirmatively for various classes of groups
  • Abelian groups: every finite abelian group is the Galois group of a regular extension of the rationals (Kummer theory)
  • Symmetric groups: every symmetric group $S_n$ is the Galois group of a regular extension of the rationals (Noether's theorem)
  • Many simple groups: techniques from algebraic geometry and representation theory have been used to construct regular extensions with Galois groups isomorphic to various simple groups
• The regular inverse Galois problem provides a stronger version of the inverse Galois problem and has important applications in algebraic geometry and arithmetic geometry

Open Problems and Challenges

• Despite significant progress, the inverse Galois problem remains open for many classes of finite groups
  • Non-solvable groups: constructing Galois extensions with prescribed non-solvable Galois groups is a major challenge
  • Groups with complex structure: groups with intricate subgroup structure or representation-theoretic properties pose difficulties for current techniques
• The inverse Galois problem for specific groups, such as the Monster group or the Mathieu groups, remains unresolved
  • Constructing Galois extensions with these groups as Galois groups requires advanced techniques from algebraic geometry, representation theory, and group theory
• The development of new methods and techniques to tackle the inverse Galois problem for challenging classes of groups is an active area of research in modern Galois theory

Implications of a Complete Solution

Characterization of Galois Groups

• A complete solution to the inverse Galois problem would provide a full characterization of the finite groups that can arise as Galois groups over the rational numbers
  • Would answer the question of which finite groups can be realized as automorphism groups of field extensions
  • Would establish a deep connection between the structure of finite groups and the Galois theory of field extensions

Encoding Groups into Polynomial Equations

• A positive solution to the inverse Galois problem would imply that every finite group can be "encoded" into a polynomial equation over the rationals
  • The Galois group of the splitting field of the polynomial would be isomorphic to the given finite group
  • This encoding would establish a profound link between group theory and field theory
  • Would allow for the study of finite groups using techniques from Galois theory and algebraic geometry

Implications for Related Problems

• A complete solution to the inverse Galois problem would have implications for related problems in Galois theory
  • The Noether problem: asks about the rationality of fixed fields under group actions
  • The Shafarevich conjecture: states that every finite solvable group is the Galois group of a Galois extension of the rationals (proved by Shafarevich)
  • The regular inverse Galois problem: asks whether every finite group appears as the Galois group of a regular extension of the rationals
• The techniques and methods developed to solve the inverse Galois problem would likely have applications in other areas of mathematics
  • Algebraic geometry: techniques from algebraic geometry (Belyi's theorem, Riemann surfaces) have been crucial in constructing Galois extensions
  • Representation theory: the regular inverse Galois problem involves the study of group representations and their realizations over the rationals
  • Number theory: the inverse Galois problem is closely related to the study of algebraic number fields and their Galois groups

Landmark Achievement in Mathematics

• The resolution of the inverse Galois problem would be a landmark achievement in modern mathematics
  • Would represent a major advance in our understanding of the structure and properties of finite groups
  • Would establish deep connections between group theory, field theory, and algebraic geometry
  • Would open up new avenues for research in Galois theory, number theory, and related areas
• A complete solution to the inverse Galois problem would be a testament to the power and depth of modern algebraic methods and would showcase the importance of interdisciplinary approaches in mathematics