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 β = , 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, 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. In fact, F is the totally real subfield of K mentioned above. This follows from Dirichlet's unit theorem.

Examples