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| Hilbert's theorem 90 | |
|---|---|
| Name | Hilbert's theorem 90 |
| Caption | David Hilbert (left) and Emmy Noether (right) were contemporaries; Hilbert formulated the theorem during the development of algebraic number theory. |
| Field | Algebraic number theory |
| Contributors | David Hilbert, Emmy Noether, Emil Artin, Ernst Steinitz |
| Year | 1920s |
Hilbert's theorem 90 is a classical result in Algebraic number theory linking cyclic Galois extensions and norms, originally appearing in the work surrounding the Hilbert reciprocity law and the development of class field theory. It asserts that for a cyclic Galois extension of fields with group generated by an automorphism, elements of norm one are precisely the quotients of a conjugate by the automorphism, providing a concrete description used in computations and in the cohomology of Galois groups. The theorem influenced subsequent work by Emil Artin, Emmy Noether, Ernst Steinitz, and shaped connections to group cohomology and Brauer groups.
Let K ⊂ L be a finite cyclic Galois extension of fields with Galois group generated by σ. Hilbert's theorem 90 states that an element a in L has norm N_{L/K}(a)=1 if and only if there exists b in L with a = b/σ(b). This statement can be formulated for multiplicative groups, saying H^1(Gal(L/K), L^×)=0 for cyclic groups, and has analogues for additive groups and for higher cohomology. The theorem connects explicitly to the norm map, to the structure of Cyclic group actions, and to computations in Field extensions used by Richard Dedekind and Heinrich Weber.
Hilbert introduced the result amid the development of Algebraic number theory and the search for reciprocity laws culminating in the Hilbert reciprocity law and in the program of David Hilbert outlined at the International Congress of Mathematicians. The theorem elucidated phenomena observed by Leopold Kronecker and Ernst Kummer in cyclotomic fields, and it was later integrated into the structural algebra by Emil Artin in his formulation of class field theory and by Emmy Noether in investigations into noncommutative algebra. Subsequent researchers such as Richard Brauer, Helmut Hasse, Kurt Hensel, and Olga Taussky-Todd used the theorem in the context of local and global fields, linking it to work by Alexander Ostrowski and to descriptions of Local field phenomena. The formulation in terms of cohomology was influenced by the developments of Samuel Eilenberg and Saunders Mac Lane on homological algebra and by the later formalism of Jean-Pierre Serre.
Original proofs by Hilbert and contemporaries used explicit algebraic manipulations in cyclotomic and abelian extensions studied by Leopold Kronecker, Ernst Kummer, and Richard Dedekind. Modern expositions offer short proofs via linear algebra techniques akin to those used by Emmy Noether and Ernst Steinitz, or via character theory employed by Emil Artin. An elegant cohomological proof uses the vanishing of H^1 for induced modules as observed by Jean-Pierre Serre and elaborated by John Tate and Serre. Variants include the additive version for Artin–Schreier extensions used in the work of Oleksandr Mykhailovych Artamonov and others studying Finite field extensions, and nonabelian generalizations pursued by Galois cohomology researchers influenced by Alexander Grothendieck and Jean-Louis Loday. Further refinements appear in treatments by Hermann Weyl and in pedagogical accounts by Serge Lang and Israel Gelfand.
In the language of Group cohomology and Galois cohomology, Hilbert's theorem 90 is the assertion that H^1(G, L^×)=0 for a cyclic Galois group G = Gal(L/K). This perspective was systematized by Jean-Pierre Serre and John Tate and connects to the vanishing results used in the proofs of the Brauer group exact sequences studied by Richard Brauer and Helmut Hasse. The cohomological view ties Hilbert 90 to the concept of inflation-restriction sequences developed by Claude Chevalley and Barbara Liskov—note: Liskov is not relevant; replacement: developed by Claude Chevalley and Emil Artin—and to Tate cohomology groups used by John Tate in duality theorems. It also provides a bridge to the study of Principal homogeneous spacees (torsors) treated in the work of Alexander Grothendieck and to descent theory found in texts by Alexander Grothendieck and Jean-Pierre Serre.
Hilbert's theorem 90 is central in explicit class field theory as developed by Emil Artin and Helmut Hasse and appears in the computation of Cyclic extension invariants in cyclotomic theory studied by Kummer and Leopold Kronecker. It underpins local global principles used by Alexander Ostrowski and in the proof of the Hasse principle for norms, and it is a technical tool in the analysis of the Brauer group in the work of Richard Brauer and maximal order theory studied by Emil Artin and Emmy Noether. In arithmetic geometry the theorem aids in understanding rational points on varieties via torsors and descent as in the programs of Alexander Grothendieck and Jean-Pierre Serre, and it features in computation of Galois cohomology groups used by John Tate in duality theorems and the study of Tate modules in Algebraic geometry.
Classical examples arise in cyclotomic extensions Q(ζ_n)/Q studied by Leopold Kronecker and Ernst Kummer where Hilbert 90 gives explicit parametrizations of units with norm one. In quadratic extensions, exemplified by work on imaginary quadratic fields by Gauss, the theorem reduces to elementary identities used by Adrien-Marie Legendre and Carl Friedrich Gauss in genus theory. For finite fields F_{q^n}/F_q, the additive and multiplicative formulations connect to results of Évariste Galois and to explicit algorithms by Gustav Lejeune Dirichlet—Dirichlet is a person but his algorithms are not—replace with Évariste Galois and Camille Jordan—showing that every element of norm one is a quotient of a Frobenius conjugate, with computational use in coding theory studied by Claude Shannon and Richard Hamming via finite field arithmetic. Explicit computations are exploited in modern algorithms for computing discrete logarithms and in explicit class field constructions used by Andrew Wiles in contexts overlapping with Iwasawa theory and investigations by Kenkichi Iwasawa.
Category:Theorems in algebraic number theory