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 and Taniyama.

Formal definition

A number field ${\displaystyle K}$ is a CM-field if it is a quadratic extension ${\displaystyle K/F}$ where the base field ${\displaystyle F}$ is totally real but ${\displaystyle K}$ is totally imaginary; i.e., every embedding of ${\displaystyle F}$ into ${\displaystyle \mathbb {C} }$ lies entirely within ${\displaystyle \mathbb {R} }$, but there is no embedding of ${\displaystyle K}$ into ${\displaystyle \mathbb {R} }$.

In other words, there is a subfield ${\displaystyle F}$ of ${\displaystyle K}$ such that ${\displaystyle K}$ is generated over ${\displaystyle F}$ by a single square root of an element, say ${\displaystyle \beta ={\sqrt {\alpha }}}$, in such a way that the minimal polynomial of ${\displaystyle \beta }$ 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, ${\displaystyle \sigma (\alpha )<0}$.

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

• 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 ${\displaystyle x^{4}+x^{3}-x^{2}-x+1}$.

Notes

References