CM-field

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.

Formal definition
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 $$\mathbb C $$ lies entirely within $$\mathbb R $$, but there is no embedding of K into $$\mathbb R $$.

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 &beta; = $$\sqrt{\alpha} $$, in such a way that the minimal polynomial of β over the rational number field $$ \mathbb Q$$ has all its roots non-real complex numbers. For this α should be chosen totally negative, so that for each embedding σ of $$F$$ into the real number field, &sigma;(&alpha;) < 0.

Properties
One feature of a CM-field is that complex conjugation on $$\mathbb C $$ induces an automorphism on the field which is independent of its embedding into $$\mathbb C$$. 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 $$\mathbb Z$$-rank as that of K. In fact, F is the totally real subfield of K mentioned above. This follows from Dirichlet's unit theorem.

Examples

 * 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 $$ \mathbb Q (\zeta_n) $$, which is generated by a primitive nth root of unity. It is a totally imaginary quadratic extension of the totally real field $$ \mathbb Q (\zeta_n +\zeta_n^{-1}). $$ The latter is the fixed field of complex conjugation, and $$ \mathbb Q (\zeta_n) $$ is obtained from it by adjoining a square root of  $$ \zeta_n^2+\zeta_n^{-2}-2 = (\zeta_n - \zeta_n^{-1})^2.  $$
 * 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($\overline{Q}$/QR) is generated (as a closed subgroup) by all elements of order 2 in Gal($\overline{Q}$/Q), and Gal($\overline{Q}$/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.
 * One example of a totally imaginary field which is not CM is the number field defined by the polynomial $$x^4 + x^3 - x^2 - x + 1$$.