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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hilbert symbol canonical | 3 |
| Hilbert reciprocity law | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871700 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hilbert symbol Context triple: [Jacobi symbol, relatedConcept, Hilbert symbol]
-
A.
Hasse invariant
The Hasse invariant is an arithmetic invariant in number theory and algebraic geometry that classifies structures such as quadratic forms or elliptic curves over local and global fields, playing a key role in local-global principles.
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B.
Hasse norm theorem
The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
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C.
Legendre symbol
The Legendre symbol is a number-theoretic function that indicates whether an integer is a quadratic residue modulo an odd prime, taking values 1, −1, or 0 accordingly.
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D.
Hilbert class field
The Hilbert class field of a number field is its maximal unramified abelian extension, central in class field theory as it corresponds to the field’s ideal class group.
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E.
Jacobi symbol
The Jacobi symbol is a number-theoretic function that generalizes the Legendre symbol and plays a key role in quadratic residues and primality testing in modular arithmetic.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hilbert symbol Target entity description: 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.
-
A.
Hasse invariant
The Hasse invariant is an arithmetic invariant in number theory and algebraic geometry that classifies structures such as quadratic forms or elliptic curves over local and global fields, playing a key role in local-global principles.
-
B.
Hasse norm theorem
The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
-
C.
Legendre symbol
The Legendre symbol is a number-theoretic function that indicates whether an integer is a quadratic residue modulo an odd prime, taking values 1, −1, or 0 accordingly.
-
D.
Hilbert class field
The Hilbert class field of a number field is its maximal unramified abelian extension, central in class field theory as it corresponds to the field’s ideal class group.
-
E.
Jacobi symbol
The Jacobi symbol is a number-theoretic function that generalizes the Legendre symbol and plays a key role in quadratic residues and primality testing in modular arithmetic.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hilbert symbol Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hilbert reciprocity law