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 $\mathbb {Q}$ 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 $\mathbb {Z}$-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 $\mathbb {Q} (\zeta _{n})$, 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 $x^{4}+x^{3}-x^{2}-x+1$.

References

• Remak, Robert (1954). "Über algebraische Zahlkörper mit schwachem Einheitsdefekt". Compositio Mathematica (in German). 12: 35–80. MR 0063403. Zbl 0055.26805.

• Shimura, Goro (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan. Vol. 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. Vol. 6. Tokyo: Mathematical Society of Japan. MR 0125113.

• Washington, Lawrence C. (1997) [1982]. Introduction to Cyclotomic Fields. Graduate Texts in Mathematics. Vol. 83 (2nd ed.). New York: Springer. ISBN 978-1-4612-7346-2. Zbl 0966.11047.