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).

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 ${\displaystyle \mathbb {C} }$ lies entirely within ${\displaystyle \mathbb {R} }$, but there is no embedding of K into ${\displaystyle \mathbb {R} }$.

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