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CM-field

Complex multiplication field

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Source updated August 3, 2026

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.

01Formal definition

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

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

02Properties

One feature of a CM-field is that complex conjugation on $\mathbb {C}$ induces an automorphism on the field which is independent of its embedding into $\mathbb {C}$ . In the notation given, it must negate $\beta$ .

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 $\mathbb {Z}$ -rank as that of $K$ . In fact, $F$ is the totally real subfield of $K$ mentioned above. This follows from Dirichlet's unit theorem.

03Examples

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 $\mathbb {Q} (\zeta _{n})$ , which is generated by a primitive $n$ th root of unity. It is a totally imaginary quadratic extension of the totally real field $\mathbb {Q} (\zeta _{n}+\zeta _{n}^{-1})$ . The latter is the fixed field of complex conjugation, and $\mathbb {Q} (\zeta _{n})$ is obtained from it by adjoining a square root of $(\zeta _{n}+\zeta _{n}^{-1})^{2}-4=(\zeta _{n}-\zeta _{n}^{-1})^{2}.$

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

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 $\mathbb {Z}$ . 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 $x^{4}+x^{3}-x^{2}-x+1$ .

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Sources and credits

This article is adapted from the Wikipedia article “CM-field”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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