Galois group
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| Galois group | |
|---|---|
| Name | Galois group |
| Introduced | Évariste Galois |
| Field | Mathematics |
| Subfield | Algebra |
| Applications | Number theory, Algebraic geometry, Cryptography |
Galois group A Galois group is a central object in modern algebra introduced by Évariste Galois that relates field extensions to group theory. It appears across Évariste Galois, Augustin-Louis Cauchy, Niels Henrik Abel, Carl Friedrich Gauss, and influenced later work by David Hilbert, Emil Artin, Émile Picard, and Emmy Noether. The concept bridges developments in Paris, Berlin, Göttingen, Cambridge (UK), and Princeton (New Jersey) mathematical traditions.
Definition and basic properties
The Galois group of a field extension captures automorphisms; foundational contributors include Évariste Galois, Évariste Galois's contemporaries, and later formalizers such as Évariste Galois's followers and Émile Picard. Key early results were refined in work at École Normale Supérieure, Collège de France, University of Göttingen, University of Berlin, and University of Cambridge. Basic properties link to theorems by Niels Henrik Abel, Carl Friedrich Gauss, David Hilbert, Emil Artin, and Emmy Noether. Important examples and invariants are connected with Frobenius (mathematician), Richard Dedekind, Leopold Kronecker, Alexander Grothendieck, and Helmut Hasse.
Galois groups of field extensions
For a finite extension investigated by Emil Artin and Emmy Noether, the Galois group is the group of field automorphisms fixing the base field; classical studies occurred in Paris, Göttingen, Hilbert's Problems, and Bourbaki circles. Important extensions studied by Carl Ludwig Siegel, André Weil, Kurt Hensel, Heinrich Weber, and Richard Brauer include cyclotomic extensions linked to Carl Friedrich Gauss and abelian class field theory developed by Erich Hecke, Teiji Takagi, Helmut Hasse, and John Tate. Nonabelian extensions feature in work by Galois theorists across institutions like Princeton University, Harvard University, University of Chicago, and Institut des Hautes Études Scientifiques.
Galois groups of polynomials
The polynomial-centric perspective, advanced by Évariste Galois, Niels Henrik Abel, Joseph-Louis Lagrange, and Pierre-Simon Laplace, studies permutations of roots; significant contributors include Arthur Cayley, William Rowan Hamilton, Sophus Lie, and Felix Klein. Classical computations of symmetric and alternating group realizations relate to Camille Jordan, Évariste Galois's legacy, and later algorithmic treatments by David Hilbert, Emil Artin, Kurt Gödel's contemporaries, and researchers at Institut Henri Poincaré. The role of discriminants and resolvents connects to work by Galois, Lagrange, Cauchy, Abel, Galois's contemporaries, and Émile Picard.
Structure and examples (abelian, cyclic, symmetric, dihedral)
Canonical examples—abelian groups studied in Kronecker and Leopold Kronecker's programs, cyclic groups tied to Gauss and Kummer, symmetric groups analyzed by Camille Jordan and Évariste Galois, and dihedral groups appearing in classical geometry—trace through contributions at École Polytechnique, University of Göttingen, Sorbonne, and Princeton. Abelian class field theory culminated in work by Erich Hecke, Teiji Takagi, Helmut Hasse, John Tate, and Claude Chevalley. Cyclic extensions appear in cyclotomic theory by Carl Friedrich Gauss and in Kummer theory by Ernst Kummer and Leopold Kronecker. Symmetric and alternating group realizations were studied by Émile Picard, Camille Jordan, Arthur Cayley, and later by Emil Artin. Dihedral examples appear in polygon constructibility problems dating back to Gauss and explored by Carl Friedrich Gauss, Niels Henrik Abel, and Pierre Wantzel.
Galois correspondence and applications
The fundamental Galois correspondence, formalized by Emil Artin and popularized by Emmy Noether and David Hilbert, establishes an inclusion-reversing bijection between intermediate fields and subgroups; this principle influenced research at University of Göttingen, Princeton University, ETH Zurich, and École Normale Supérieure. Applications touch major results and conjectures addressed by Andrew Wiles, Gerd Faltings, Pierre Deligne, Alexander Grothendieck, and Jean-Pierre Serre—notably in proofs of cases of the Taniyama–Shimura conjecture, Fermat's Last Theorem, and in the arithmetic of elliptic curves studied at Institute for Advanced Study. Links to algebraic number theory involve Richard Dedekind, Erich Hecke, Teiji Takagi, Helmut Hasse, and John Tate.
Computation and algorithms
Algorithmic computation of Galois groups was advanced by computational algebraists at Massachusetts Institute of Technology, University of Cambridge, University of California, Berkeley, and Princeton University. Key contributors include Richard Dedekind historically, and modern developers like John H. Conway's circle, Henri Cohen, János Kollár's networks, and teams behind SageMath, GAP (software), Magma (software), and PARI/GP. Algorithmic methods incorporate work by Emil Artin, David Hilbert, André Weil, Michael F. Singer, Mark van Hoeij, and researchers at INRIA and Microsoft Research.
Advanced topics (profinite groups, inverse Galois problem)
Advanced theory connects Galois groups to profinite groups studied by André Weil, Jean-Pierre Serre, Alexander Grothendieck, Saul Kripke's era contemporaries in set theory, and to the Inverse Galois Problem pursued by Hilbert and modern researchers like Malle, Boston (mathematician), Taylor (mathematician), Richard Taylor, Michael Fried, David Harbater, Jean-Pierre Serre, and Hans-Werner Lenstra Jr.. The absolute Galois group of Q features centrally in Grothendieck's anabelian geometry and in the Langlands program propelled by Robert Langlands, Pierre Deligne, Michael Harris, Richard Taylor, and Andrew Wiles. Profound interactions involve Iwasawa theory developed by Kenkichi Iwasawa, Barry Mazur, John Coates, and Mazur's collaborators at institutions like Harvard University and Cambridge (UK).