A CM-field, short for "Complex Multiplication field," is a type of number field that is significant in algebraic number theory, particularly in the study of elliptic curves and modular forms. More specifically, a CM-field is an imaginary quadratic field \(K\) that arises from the theory of elliptic curves with complex multiplication by a certain ring of integers.
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A CM-field is a totally imaginary number field that is a quadratic extension of a totally real number field. Its nontrivial automorphism over that real subfield acts as complex conjugation in every complex embedding. Imaginary quadratic number fields are examples. An arbitrary totally imaginary number field need not have such a real subfield of index two.