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 formula_1 lies entirely within formula_2, but there is no embedding of "K" into formula_2.
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 = formula_4,
in such a way that the minimal polynomial of β over the rational number field formula_5 has all its roots non-real complex numbers. For this α should be chosen "totally negative", so that for each embedding σ of formula_6 into the real number field,
σ(α) < 0.
Properties.
One feature of a CM-field is that complex conjugation on formula_1 induces an automorphism on the field which is independent of its embedding into formula_8. 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 formula_9-rank as that of "K" . In fact, "F" is the totally real subfield of "K" mentioned above. This follows from Dirichlet's unit theorem.