Hilbert symbol
E697755
The Hilbert symbol is a local arithmetic invariant in number theory that encodes whether a quadratic form represents zero over a given local field, playing a central role in local class field theory and reciprocity laws.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Hilbert reciprocity law | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
invariant
ⓘ
local invariant ⓘ mathematical concept ⓘ number theoretic invariant ⓘ |
| alsoKnownAs | Hilbert norm residue symbol ⓘ |
| appearsIn |
class field theory of local number fields
ⓘ
explicit formulas for local reciprocity maps ⓘ theory of central simple algebras as symbol algebras ⓘ |
| centralIn |
local class field theory
ⓘ
local reciprocity laws ⓘ |
| codomain | multiplicative group {±1} ⓘ |
| definedFor | pairs of elements of a local field modulo squares ⓘ |
| domain |
completions of number fields
ⓘ
local fields ⓘ p-adic fields ⓘ |
| encodes | whether a quadratic form represents zero over a local field ⓘ |
| field | number theory ⓘ |
| generalizationOf |
Legendre symbol
NERFINISHED
ⓘ
quadratic residue symbol ⓘ |
| historicalPeriod | early 20th century mathematics ⓘ |
| mathematicalArea |
algebra
ⓘ
arithmetic geometry ⓘ class field theory ⓘ |
| namedAfter | David Hilbert NERFINISHED ⓘ |
| property |
bilinear in each argument modulo squares
ⓘ
continuous with respect to the local field topology ⓘ symmetric in its two arguments for local fields of characteristic not 2 ⓘ takes values in {1,-1} ⓘ trivial on pairs of local norms ⓘ |
| relatedTo |
Brauer group
NERFINISHED
ⓘ
Galois cohomology ⓘ Hilbert reciprocity law NERFINISHED ⓘ Kummer theory NERFINISHED ⓘ cup product in Galois cohomology ⓘ norm residue map ⓘ |
| role |
local arithmetic invariant
ⓘ
measures local norm residue information ⓘ measures local solvability of quadratic equations ⓘ |
| satisfies | product formula over all completions of a global field ⓘ |
| subfield |
algebraic number theory
ⓘ
local class field theory NERFINISHED ⓘ |
| typicalNotation |
(a,b)_K
ⓘ
(a,b)_v ⓘ |
| usedIn |
classification of central simple algebras over local fields
ⓘ
description of the Brauer group of local fields ⓘ formulation of the product formula for local invariants ⓘ proofs of quadratic reciprocity ⓘ study of quadratic forms over local fields ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hilbert reciprocity law