CM-field

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 ${\displaystyle \mathbb {R} }$.

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 = ${\displaystyle {\sqrt {\alpha }}}$, in such a way that the minimal polynomial of β over the rational number field ${\displaystyle \mathbb {Q} }$ has all its roots non-real complex numbers. For this α should be chosen totally negative, so that for each embedding σ of ${\displaystyle F}$ into the real number field, σ(α) < 0.

One feature of a CM-field is that complex conjugation on ${\displaystyle \mathbb {C} }$ induces an automorphism on the field which is independent of its embedding into ${\displaystyle \mathbb {C} }$. 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 ${\displaystyle \mathbb {Z} }$-rank as that of K (Remak 1954). In fact, F is the totally real subfield of K mentioned above. This follows from Dirichlet's unit theorem.

• The simplest, and motivating, example of a CM-field is an imaginary quadratic field, for which the totally real subfield is just the field of rationals.
• One of the most important examples of a CM-field is the cyclotomic field ${\displaystyle \mathbb {Q} (\zeta _{n})}$, which is generated by a primitive nth root of unity. It is a totally imaginary quadratic extension of the totally real field ${\displaystyle \mathbb {Q} (\zeta _{n}+\zeta _{n}^{-1}).}$ The latter is the fixed field of complex conjugation, and ${\displaystyle \mathbb {Q} (\zeta _{n})}$ is obtained from it by adjoining a square root of ${\displaystyle \zeta _{n}^{2}+\zeta _{n}^{-2}-2=(\zeta _{n}-\zeta _{n}^{-1})^{2}.}$
• The union Q^CM of all CM fields is similar to a CM field except that it has infinite degree. It is a quadratic extension of the union of all totally real fields Q^R. The absolute Galois group Gal(Q/Q^R) is generated (as a closed subgroup) by all elements of order 2 in Gal(Q/Q), and Gal(Q/Q^CM) is a subgroup of index 2. The Galois group Gal(Q^CM/Q) has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group Gal(Q^R/Q).
• If V is a complex abelian variety of dimension n, then any abelian algebra F of endomorphisms of V has rank at most 2n over Z. If it has rank 2n and V is simple then F is an order in a CM-field. Conversely any CM field arises like this from some simple complex abelian variety, unique up to isogeny.
• One example of a totally imaginary field which is not CM is the number field defined by the polynomial ${\displaystyle x^{4}+x^{3}-x^{2}-x+1}$.

• Remak, Robert (1954), "Über algebraische Zahlkörper mit schwachem Einheitsdefekt", Compositio Mathematica (in German), 12: 35–80, Zbl 0055.26805
• Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton, N.J.: Princeton University Press
• Shimura, Goro; Taniyama, Yutaka (1961), Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Tokyo: The Mathematical Society of Japan, MR 0125113
• Washington, Lawrence C. (1996). Introduction to Cyclotomic fields (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0. Zbl 0966.11047.