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 $K$ 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 $\mathbb{\mathbb{Q}}$ 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, $\sigma (\alpha )<0$.

Properties

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

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 $\mathbb{\mathbb{Q}}({\zeta }_n)$, which is generated by a primitive $n$th root of unity. It is a totally imaginary quadratic extension of the totally real field $\mathbb{\mathbb{Q}}({\zeta }_n+{\zeta }_n^{-1})$. The latter is the fixed field of complex conjugation, and $\mathbb{\mathbb{Q}}({\zeta }_n)$ 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 ${\mathbb{\mathbb{Q}}}^{\mathrm{C}\mathrm{M}}$ 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 ${\mathbb{\mathbb{Q}}}^{\mathrm{R}}$. The absolute Galois group $\mathrm{Gal}(\overline{\mathbb{\mathbb{Q}}}/{\mathbb{\mathbb{Q}}}^{\mathrm{R}})$ is generated (as a closed subgroup) by all elements of order 2 in $\mathrm{Gal}(\overline{\mathbb{\mathbb{Q}}}/\mathbb{\mathbb{Q}})$, and $\mathrm{Gal}(\overline{\mathbb{\mathbb{Q}}}/{\mathbb{\mathbb{Q}}}^{\mathrm{C}\mathrm{M}})$ is a subgroup of index 2. The Galois group $\mathrm{Gal}({\mathbb{\mathbb{Q}}}^{\mathrm{C}\mathrm{M}}/\mathbb{\mathbb{Q}})$ has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group $\mathrm{Gal}({\mathbb{\mathbb{Q}}}^{\mathrm{R}}/\mathbb{\mathbb{Q}})$.

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

References

• Remak, Robert (1954). "Über algebraische Zahlkörper mit schwachem Einheitsdefekt" (in German). Compositio Mathematica 12: 35–80. 

• Shimura, Goro (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan. 11. Princeton University Press. ISBN 978-0-69108092-5. 

• Shimura, Goro; Taniyama, Yutaka (1961). Complex Multiplication of Abelian Varieties and Its Applications to Number Theory. Publications of the Mathematical Society of Japan. 6. Tokyo: Mathematical Society of Japan. 

• Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields. Graduate Texts in Mathematics. 83 (2nd ed.). New York: Springer. ISBN 978-1-4612-7346-2.