Hilbert symbol

Hilbert symbol
Name	Hilbert symbol
Field	Number theory
Introduced	1928
Introduced by	David Hilbert
Related	Local class field theory, Quadratic forms, Hilbert reciprocity, Artin map, Brauer group

Hilbert symbol is a bilinear form-like pairing arising in algebraic number theory that encodes local reciprocity and quadratic residue information for local fields such as p-adic number, real number, and finite extension fields. It connects objects from David Hilbert's program to later advances by Emil Artin, Helmut Hasse, John Tate, and Claude Chevalley and plays a central role in the development of local class field theory, global class field theory, and the structure of the Brauer group of a field. The symbol provides a compact algebraic invariant implemented in the study of quadratic forms, Galois cohomology, and explicit reciprocity laws used by researchers working on conjectures influenced by André Weil, Alexander Grothendieck, and Jean-Pierre Serre.

Definition

For a local field K with multiplicative group K^× and integer n dividing the order of certain cyclotomic extensions, the Hilbert symbol is a pairing (·,·)_K: K^× × K^× → μ_n where μ_n denotes the group of n-th roots of unity in a fixed algebraic closure. The construction was formalized following ideas of David Hilbert and was developed concretely by Helmut Hasse and Emil Artin in the context of norm residue maps and the Artin reciprocity law. In the quadratic case (n=2) for a non-archimedean local field or the real field, the pairing takes values in {±1} and encapsulates whether certain binary quadratic forms represent zero nontrivially, a perspective tied to work by Adolf Hurwitz and Friedrich Hirzebruch in the classification of forms.

Properties

The Hilbert symbol satisfies alternating and bilinear-like relations reminiscent of cup product properties in Galois cohomology and matches the behavior of the Artin map under local reciprocity. Key properties include: - Bimultiplicativity: (ab,c)_K = (a,c)_K (b,c)_K and (a,bc)_K = (a,b)_K (a,c)_K, reflecting multiplicative behavior found in results of Emil Artin and John Tate. - Nondegeneracy: For fixed a ∈ K^×, the map b ↦ (a,b)_K is trivial only when a lies in norms from certain cyclic extensions characterized in the work of Helmut Hasse and Shafarevich. - Symmetry relations: In quadratic situations the symbol is symmetric or skew-symmetric depending on parity, echoing patterns appearing in studies by Erich Hecke and Carl Ludwig Siegel. - Compatibility with valuations: For discretely valued fields the symbol interacts predictably with uniformizers and residue fields, an insight used by Kurt Hensel and Richard Dedekind in local field analysis. - Reciprocity: The global product formula, known as Hilbert reciprocity, asserts that the product of local Hilbert symbols over all completions of a global field equals 1; this was proved by David Hilbert and placed in the broader context of global class field theory by Emil Artin and Teiji Takagi.

Computation and formulas

Explicit computation uses residue maps, norm criteria, and valuation data, often reducing problems to residue field computations studied by Heinrich Weber and Heinrich Martin Weber. Techniques include: - Local reciprocity via the Artin map: computation of (a,b)_K via the image of b under the local Artin map in cyclic extensions generated by n-th roots of a, aligning with methods of Emil Artin and Helmut Hasse. - Valuation-based formulae: for a discretely valued field with uniformizer π and units u, v, one reduces (π^r u, π^s v)_K to explicit powers involving r, s and the residue symbol in the residue field, following approaches pioneered by Kurt Hensel and applied by Jean-Pierre Serre. - Explicit quadratic formulas: for n=2 over local fields with odd residual characteristic, the Hilbert symbol can be computed by Legendre and Jacobi-type symbols extending work of Adrien-Marie Legendre and Carl Gustav Jacobi and refined by Helmut Hasse. - Topological methods: in the archimedean case computations reduce to sign considerations for real number fields or triviality for complex number fields, as noted in classical analysis by Bernhard Riemann.

Examples and special cases

- Real field: For K = real number, the quadratic Hilbert symbol is (a,b)_R = −1 precisely when both a and b are negative, a fact connected to the theory of Euler and Gauss on signs and quadratic forms. - Complex field: For K = complex number the Hilbert symbol is trivial for all elements, consistent with complex fields being algebraically closed, a situation tied to properties studied by Gotthold Eisenstein. - p-adic fields: For K = Q_p, explicit tables compute (a,b)_{Q_p} via valuation and residue class computations; key calculations appear in the work of Kurt Hensel, Helmut Hasse, and modern expositions by Jürgen Neukirch and Serge Lang. - Finite extensions: For finite extensions of Q_p or function fields over finite fields, the symbol governs the splitting behavior of quaternion algebras and cyclic algebras studied by Richard Brauer and Claude Chevalley.

Relationship with local and global class field theory

The Hilbert symbol provides a concrete incarnation of the local norm residue symbol and is intertwined with the Artin reciprocity law and the description of the local reciprocity map in local class field theory. Locally, it identifies cyclic extensions via norm criteria central to work by Helmut Hasse and Emil Artin; globally, the product formula—Hilbert reciprocity—matches the reciprocity maps across all completions of a global field, forming an essential ingredient in proofs by Teiji Takagi and modern treatments by John Tate and Jean-Pierre Serre studying the arithmetic of number fields and function fields.

Generalizations and higher-degree symbols

Generalizations replace μ_2 by μ_n and extend the pairing to higher Milnor K-theory and Galois cohomology groups H^i(·,μ_n), building on ideas by Alexander Merkurjev, Andrei Suslin, and Vladimir Voevodsky in the proof of the Bloch–Kato conjecture and the norm residue isomorphism. Higher-degree norm residue symbols and cup products generalize classical Hilbert symbol behavior to link cyclic algebras, the Brauer group, and motivic cohomology, themes pursued by Jean-Louis Colliot-Thélène, Kazuya Kato, and Spencer Bloch in modern arithmetic geometry.

Category:Number theory