cyclotomic fields
E157383
Cyclotomic fields are number fields obtained by adjoining complex roots of unity to the rationals, playing a central role in algebraic number theory and classical geometric constructibility.
All labels observed (1)
| Label | Occurrences |
|---|---|
| cyclotomic fields canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382381 — 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: cyclotomic fields Context triple: [construction of the regular 17-gon with straightedge and compass, isLinkedTo, cyclotomic fields]
-
A.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
-
B.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
C.
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.
-
D.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
E.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
- 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: cyclotomic fields Target entity description: Cyclotomic fields are number fields obtained by adjoining complex roots of unity to the rationals, playing a central role in algebraic number theory and classical geometric constructibility.
-
A.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
-
B.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
C.
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.
-
D.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
E.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic number field
ⓘ
mathematical object ⓘ number field ⓘ |
| appliedIn |
construction problems in classical geometry
ⓘ
study of Fermat’s Last Theorem (classical approach) ⓘ study of regular and irregular primes ⓘ |
| baseField | rational numbers ⓘ |
| centralTo |
Iwasawa theory
ⓘ
Kummer theory ⓘ class field theory over Q ⓘ theory of cyclotomic integers ⓘ |
| characterizedBy |
every finite abelian extension of Q is contained in a cyclotomic field
ⓘ
regular n-gon is constructible with straightedge and compass iff Q(ζ_n) has degree a power of 2 ⓘ |
| constructedBy | adjoining a primitive n-th root of unity ζ_n to Q ⓘ |
| contains | n-th roots of unity ⓘ |
| definedAs | a number field obtained by adjoining a primitive root of unity to the rational numbers ⓘ |
| degreeOverQ | φ(n) ⓘ |
| fieldOfStudy |
algebraic number theory
ⓘ
number theory ⓘ |
| generatedAsQAlgebraBy | root of Φ_n(x) ⓘ |
| generatedBy | primitive n-th root of unity ⓘ |
| hasAssociatedLFunction | Dirichlet L-functions attached to characters of (Z/nZ)× ⓘ |
| hasAssociatedPolynomial | n-th cyclotomic polynomial Φ_n(x) ⓘ |
| hasClassGroup | ideal class group depending on n ⓘ |
| hasDiscriminant | explicit formula in terms of n and its prime factors ⓘ |
| hasEmbedding | embeds into C via complex embeddings ⓘ |
| hasGaloisGroup | multiplicative group of units modulo n ⓘ |
| hasGaloisGroupIsomorphicTo | (Z/nZ)× ⓘ |
| hasInvariant | Euler totient φ(n) as its degree over Q ⓘ |
| hasMinimalPolynomialOverQ | Φ_n(x) ⓘ |
| hasPrimeDecomposition | rational primes factor according to congruence conditions modulo n ⓘ |
| hasProperty |
abelian extension of Q
ⓘ
class number may exceed 1 ⓘ finite extension of Q ⓘ normal extension of Q ⓘ separable extension of Q ⓘ totally imaginary for n>2 ⓘ |
| hasRamifiedPrimes | primes dividing n ⓘ |
| hasRingOfIntegers | Z[ζ_n] for n>2 ⓘ |
| hasSubfield |
maximal real subfield Q(ζ_n+ζ_n^{-1})
ⓘ
totally real maximal real subfield ⓘ |
| hasUnitGroup | Dirichlet unit group of rank φ(n)/2−1 for n>2 ⓘ |
| hasUnramifiedPrimes | primes not dividing n ⓘ |
| isGaloisExtensionOf | Q ⓘ |
| relatedTo |
Kronecker–Weber theorem
ⓘ
constructible regular polygons ⓘ |
| studiedBy |
Ernst Eduard Kummer
ⓘ
surface form:
Ernst Kummer
Leopold Kronecker ⓘ Richard Dedekind ⓘ |
| symbolicallyDenotedAs | Q(ζ_n) ⓘ |
| usedInProofOf | Kronecker–Weber theorem ⓘ |
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: cyclotomic fields Description of subject: Cyclotomic fields are number fields obtained by adjoining complex roots of unity to the rationals, playing a central role in algebraic number theory and classical geometric constructibility.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.