CM fields
E839484
CM fields are a special class of number fields with complex multiplication structure that play a central role in algebraic number theory and the explicit class field theory envisioned by Hilbert’s twelfth problem.
All labels observed (1)
| Label | Occurrences |
|---|---|
| CM fields canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10061995 — 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: CM fields Context triple: [Hilbert’s twelfth problem, partiallySolvedFor, CM fields]
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A.
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.
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B.
cyclotomic fields
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.
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C.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
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D.
global class field theory
Global class field theory is a branch of algebraic number theory that classifies finite abelian extensions of global fields (such as number fields) in terms of their arithmetic data, particularly via idele class groups and reciprocity maps.
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E.
Kummer theory
Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
- 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: CM fields Target entity description: CM fields are a special class of number fields with complex multiplication structure that play a central role in algebraic number theory and the explicit class field theory envisioned by Hilbert’s twelfth problem.
-
A.
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.
-
B.
cyclotomic fields
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.
-
C.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
-
D.
global class field theory
Global class field theory is a branch of algebraic number theory that classifies finite abelian extensions of global fields (such as number fields) in terms of their arithmetic data, particularly via idele class groups and reciprocity maps.
-
E.
Kummer theory
Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic number theory concept
ⓘ
number field ⓘ |
| centralRoleIn |
Kronecker–Weber type results for imaginary quadratic fields
ⓘ
generalizations of the Kronecker Jugendtraum ⓘ |
| hasAutomorphism | complex conjugation ⓘ |
| hasDefinition | a CM field is a totally imaginary quadratic extension of a totally real number field ⓘ |
| hasExample |
cyclotomic field Q(zeta_n) for n>2
ⓘ
imaginary quadratic field ⓘ reflex field of an abelian variety with complex multiplication ⓘ |
| hasInvariant |
CM type
ⓘ
reflex field ⓘ |
| hasProperty |
Galois group over its maximal totally real subfield is of order 2
ⓘ
admits a theory of complex multiplication for abelian varieties defined over it ⓘ admits complex conjugation as an automorphism of order 2 ⓘ can be described by a CM type, a set of embeddings into C ⓘ can be embedded into C but not into R ⓘ contains a totally real subfield of index 2 ⓘ degree over Q is even ⓘ embeddings into C come in complex conjugate pairs ⓘ every embedding into C has image not contained in R ⓘ has a class group ⓘ has a unit group of finite rank ⓘ has a well-defined discriminant ⓘ has no real embeddings ⓘ is a finite extension of the rational numbers Q ⓘ is a quadratic extension of a totally real field ⓘ is a totally imaginary number field ⓘ is often constructed as a compositum of an imaginary quadratic field with a totally real field ⓘ is stable under complex conjugation inside C ⓘ its CM type determines associated abelian varieties up to isogeny in CM theory ⓘ its Galois closure over Q is a CM field if the field is CM ⓘ its degree over Q equals twice the degree of its maximal totally real subfield ⓘ its ring of integers is a Dedekind domain ⓘ its signature is (0, n) for some n ⓘ often appears as endomorphism algebra of CM abelian varieties ⓘ plays a key role in the theory of complex multiplication of elliptic curves ⓘ plays a key role in the theory of complex multiplication of higher-dimensional abelian varieties ⓘ special values of its Hecke L-functions generate abelian extensions in CM theory ⓘ the fixed field of complex conjugation is totally real ⓘ |
| hasSubfield |
maximal totally real subfield
ⓘ
totally real subfield of index 2 ⓘ |
| relatedTo |
CM abelian varieties
ⓘ
Grössencharacters ⓘ Hecke characters ⓘ Shimura varieties ⓘ |
| usedIn |
Hilbert twelfth problem
ⓘ
linked to:
Hilbert’s twelfth problem
complex multiplication theory ⓘ construction of abelian extensions of number fields ⓘ explicit class field theory ⓘ theory of abelian varieties with complex multiplication ⓘ |
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: CM fields Description of subject: CM fields are a special class of number fields with complex multiplication structure that play a central role in algebraic number theory and the explicit class field theory envisioned by Hilbert’s twelfth problem.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.