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 lies entirely within, but there is no embedding of K into.
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 has all its roots non-real complex numbers. For this α should be chosen totally negative, so that for each embedding σ of into the real number field,
σ < 0.

Properties

One feature of a CM-field is that complex conjugation on induces an automorphism on the field which is independent of its embedding into. 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 -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