CM-field

Properties

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 negate ${\displaystyle \beta }$.

A number field ${\displaystyle K}$ is a CM-field if and only if it has a "units defect", i.e. if it contains a proper subfield ${\displaystyle F}$ whose unit group has the same ${\displaystyle \mathbb{Z}}$-rank as that of ${\displaystyle K}$. In fact, ${\displaystyle F}$ is the totally real subfield of ${\displaystyle K}$ mentioned above. This follows from Dirichlet's unit theorem.

Examples

• 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 ${\displaystyle n}$th 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 ${\displaystyle ({\zeta }_n+{\zeta }_n^{-1}{)}^2-4=({\zeta }_n-{\zeta }_n^{-1}{)}^2.}$

• The union ${\displaystyle {\mathbb{Q}}^{\mathrm{C}\mathrm{M}}}$ 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 ${\displaystyle {\mathbb{Q}}^{\mathrm{R}}}$. The absolute Galois group ${\displaystyle \mathrm{Gal}(\overline{\mathbb{Q}}/{\mathbb{Q}}^{\mathrm{R}})}$ is generated (as a closed subgroup) by all elements of order 2 in ${\displaystyle \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$, and ${\displaystyle \mathrm{Gal}(\overline{\mathbb{Q}}/{\mathbb{Q}}^{\mathrm{C}\mathrm{M}})}$ is a subgroup of index 2. The Galois group ${\displaystyle \mathrm{Gal}({\mathbb{Q}}^{\mathrm{C}\mathrm{M}}/\mathbb{Q})}$ has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group ${\displaystyle \mathrm{Gal}({\mathbb{Q}}^{\mathrm{R}}/\mathbb{Q})}$.

• If ${\displaystyle V}$ is a complex abelian variety of dimension ${\displaystyle n}$, then any abelian algebra ${\displaystyle F}$ of endomorphisms of ${\displaystyle V}$ has rank at most ${\displaystyle 2n}$ over ${\displaystyle \mathbb{Z}}$. If it has rank ${\displaystyle 2n}$ and ${\displaystyle V}$ is simple then ${\displaystyle F}$ 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 ${\displaystyle x^4+x^3-x^2-x+1}$.