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| Galois extensions | |
|---|---|
| Name | Galois extensions |
| Field | Algebra, Number Theory |
| Introduced | Évariste Galois |
| Related | Galois group, Field extension, Splitting field |
Galois extensions
A Galois extension is a central notion in algebra connecting field extensions to group theory via the symmetry of roots, originating in the work of Évariste Galois and developed by mathematicians such as Camille Jordan, Richard Dedekind, and Emil Artin. It provides a bridge between concrete polynomial equations studied by Niels Henrik Abel and Joseph-Louis Lagrange and abstract structural results found in David Hilbert's and Emmy Noether's programs. The theory underpins major developments in algebraic number theory by figures like André Weil, Helmut Hasse, and John Tate.
A finite extension L/K is Galois when L is both normal and separable over K, a notion formalized by Emil Artin in the context of Dedekind domains and Hilbert's reciprocity, with later generalizations by Claude Chevalley and Alexander Grothendieck. Key equivalent conditions involve the fixed field of the automorphism group Aut(L/K), the equality of degree formulas appearing in Ernst Steinitz's structure theory, and the description of splitting fields for polynomials studied by Augustin-Louis Cauchy and Joseph Fourier. For infinite extensions one uses profinite groups and inverse limits as in the work of Helmut Hasse and Jean-Pierre Serre, connecting to the absolute Galois group considered by Alexander Grothendieck and Jean-Louis Verdier.
Classical examples include cyclotomic extensions Q(ζ_n)/Q analyzed by Leopold Kronecker and Richard Dedekind, quadratic extensions related to Carl Friedrich Gauss, and finite fields F_{p^n}/F_p considered by Évariste Galois and later by Emil Artin. Splitting fields of separable polynomials studied by Évariste Galois and Joseph-Louis Lagrange produce Galois extensions, while Kummer extensions appear in Ernst Kummer's and Heinrich Weber's work on Fermat's Last Theorem and ideal theory. Nontrivial geometric examples arise from function fields of algebraic curves over finite fields treated by André Weil and Alexander Grothendieck.
The group Gal(L/K) of field automorphisms is central, following Évariste Galois's insight and later formalism by Camille Jordan and Otto Schreier; its structure links to permutation groups studied by Jordan and group cohomology developed by Claude Chevalley and Jean-Pierre Serre. The Fundamental Theorem of Galois Theory establishes a bijection between intermediate fields and subgroups of Gal(L/K), a paradigm employed by David Hilbert in his XIIth problem and extended by Emil Artin in class field theory with contributions from Helmut Hasse. For infinite extensions the correspondence uses closed subgroups in the profinite topology, an approach advanced by Jean-Pierre Serre and Alexander Grothendieck in étale cohomology.
The classical problem of ruler-and-compass constructibility, addressed by Carl Friedrich Gauss in his work on the constructibility of the regular 17-gon and later by Pierre Wantzel, is characterized by Galois groups being iterated cyclic 2-extensions, a criterion refined by Niels Henrik Abel and Évariste Galois in their proofs of insolvability of the general quintic. Solvability by radicals corresponds to solvable Galois groups, a concept elaborated by Camille Jordan and Otto Hölder and used in the solution of special polynomial families studied by Évariste Galois and Émile Picard.
Compositions, intersections, and normal closures of extensions reflect closure properties explored in the algebraic frameworks of Ernst Steinitz and Emmy Noether; separable closures and algebraic closures tie into work of Leopold Kronecker and Emil Artin. The behavior under base change and tensor products is studied in the contexts developed by Alexander Grothendieck's scheme theory and Jean-Pierre Serre's Galois cohomology, while descent and extension of scalars are themes in Jean-Louis Verdier's and Pierre Deligne's contributions to algebraic geometry.
In algebraic number theory, ramification theory for primes in extensions L/K, including decomposition and inertia groups, was developed by Richard Dedekind, Leopold Kronecker, and Helmut Hasse and later refined by Emil Artin in his reciprocity law. Inertia and wild ramification play roles in local field analysis by John Tate and Serge Lang, while higher ramification groups were studied by Jean-Pierre Serre and Alexander Grothendieck in arithmetic geometry. These notions underpin class field theory advanced by Helmut Hasse, Emil Artin, and Claude Chevalley and appear in the proofs of global results by Andrew Wiles and Gerhard Frey in the context of modularity.
Galois theory connects to algebraic geometry via Alexander Grothendieck's étale fundamental group, to arithmetic geometry in the work of André Weil and Jean-Pierre Serre, and to class field theory shaped by Emil Artin and Helmut Hasse. It informs the theory of motives studied by Pierre Deligne, modular forms in the work of Srinivasa Ramanujan and Andrew Wiles, and cryptography drawing on finite field structures as in Ron Rivest's and Adi Shamir's applications. Computational aspects relate to algorithmic algebra developed by Stephen A. Cook and Richard M. Karp in complexity theory, and to explicit methods used by Henri Cohen and John Cremona in computational number theory.
Category:Field extensions