Gaussian periods
E157384
Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gaussian periods canonical | 1 |
| Kummer sums | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382383 — 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: Gaussian periods Context triple: [construction of the regular 17-gon with straightedge and compass, isLinkedTo, Gaussian periods]
-
A.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
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B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
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C.
Gaussian integers
Gaussian integers are complex numbers whose real and imaginary parts are both integers, forming a lattice in the complex plane with important applications in number theory and algebra.
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D.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
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E.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
- 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: Gaussian periods Target entity description: Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
-
A.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
Gaussian integers
Gaussian integers are complex numbers whose real and imaginary parts are both integers, forming a lattice in the complex plane with important applications in number theory and algebra.
-
D.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
-
E.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
number theoretic object ⓘ |
| appearsIn | Disquisitiones Arithmeticae ⓘ |
| definedAs | sums of roots of unity over cosets of a subgroup of the multiplicative group modulo a prime ⓘ |
| developedBy | Carl Friedrich Gauss ⓘ |
| field | number theory ⓘ |
| generalizationOf | quadratic Gauss sums ⓘ |
| hasProperty |
algebraic integers
ⓘ
closed under Galois conjugation ⓘ generate subfields of cyclotomic fields ⓘ linearly independent over the rationals in typical constructions ⓘ satisfy polynomial equations with integer coefficients ⓘ their conjugates are obtained by multiplying indices by units modulo the defining modulus ⓘ their minimal polynomials often have relatively small coefficients ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| parameterizedBy |
a modulus, typically a prime p
ⓘ
a subgroup of the multiplicative group modulo the modulus ⓘ cosets of that subgroup ⓘ |
| relatedTo |
Dirichlet characters
ⓘ
Galois theory ⓘ Gauss periods in coding theory ⓘ Gaussian periods of type (k,n) ⓘ Hilbert class field ⓘ
surface form:
Hilbert class fields of imaginary quadratic fields
Kummer theory ⓘ Weber functions and modular invariants in some class field constructions ⓘ constructible polygons ⓘ cyclotomic fields ⓘ cyclotomic polynomials ⓘ cyclotomic units ⓘ finite fields ⓘ regular 17-gon ⓘ roots of unity ⓘ |
| specialCaseOf | periods in algebraic number theory ⓘ |
| usedFor |
analyzing distribution of residues modulo primes
ⓘ
constructing certain cyclic difference sets ⓘ constructing normal bases ⓘ constructing normal bases of finite field extensions ⓘ describing subfields of cyclotomic fields ⓘ evaluating Gauss sums ⓘ explicit class field theory over Q ⓘ explicit construction of regular polygons ⓘ explicit description of intermediate fields in cyclotomic extensions ⓘ explicit generators of abelian extensions of Q ⓘ straightedge and compass constructions ⓘ studying higher power residues ⓘ studying quadratic residues ⓘ |
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: Gaussian periods Description of subject: Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Kummer sums