In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field.
The abbreviation "CM" was introduced by (Shimura & Taniyama 1961).
A number field K is a CM-field if it is a quadratic extension K/F where the base field F is totally real but K is totally imaginary. I.e., every embedding of F into
C
lies entirely within
R
, but there is no embedding of K into
R
.
In other words, there is a subfield F of K such that K is generated over F by a single square root of an element, say β =
α
, in such a way that the minimal polynomial of β over the rational number field
Q
has all its roots non-real complex numbers. For this α should be chosen totally negative, so that for each embedding σ of
K
′
into the real number field, σ(α) < 0.
One feature of a CM-field is that complex conjugation on
C
induces an automorphism on the field which is independent of its embedding into
C
. In the notation given, it must change the sign of β.
A number field K is a CM-field if and only if it has a "units defect", i.e. if it contains a proper subfield F whose unit group has the same
Z
-rank as that of K (Remak 1954). In fact, F is the totally real subfield of K mentioned above. This follows from Dirichlet's unit theorem.
The simplest, and motivating, example of a CM-field is an imaginary quadratic field, for which the totally real subfield is just the field of rationals.
One of the most important examples of a CM-field is the cyclotomic field
Q
(
ζ
n
)
, which is generated by a primitive nth root of unity. It is a totally imaginary quadratic extension of the totally real field
Q
(
ζ
n
+
ζ
n
−
1
)
.
The latter is the fixed field of complex conjugation, and
Q
(
ζ
n
)
is obtained from it by adjoining a square root of
ζ
n
2
+
ζ
n
−
2
−
2
=
(
ζ
n
−
ζ
n
−
1
)
2
.
The union QCM of all CM fields is similar to a CM field except that it has infinite degree. It is a quadratic extension of the union of all totally real fields QR. The absolute Galois group Gal(Q/QR) is generated (as a closed subgroup) by all elements of order 2 in Gal(Q/Q), and Gal(Q/QCM) is a subgroup of index 2. The Galois group Gal(QCM/Q) has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group Gal(QR/Q).
If V is a complex abelian variety of dimension n, then any abelian algebra F of endomorphisms of V has rank at most 2n over Z. If it has rank 2n and V is simple then F is an order in a CM-field. Conversely any CM field arises like this from some simple complex abelian variety, unique up to isogeny.
