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 (Shimura & Taniyama 1961).

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 β = $\sqrt{\alpha}$, 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 $K'$ into the real number field, σ(α) < 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 (Remak 1954). 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 Q^CM 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 Q^R. The absolute Galois group Gal(Q/Q^R) is generated (as a closed subgroup) by all elements of order 2 in Gal(Q/Q), and Gal(Q/Q^CM) is a subgroup of index 2. The Galois group Gal(Q^CM/Q) has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group Gal(Q^R/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.