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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| CM field | 0 |
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 NERFINISHED ⓘ |
| usedIn |
Hilbert twelfth problem
NERFINISHED
ⓘ
complex multiplication theory ⓘ construction of abelian extensions of number fields ⓘ explicit class field theory ⓘ theory of abelian varieties with complex multiplication ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.