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 .

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
β = $\sqrt$,
in such a way that the minimal polynomial of β 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,
σ(α) < 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 change the sign of β.

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 nth root of unity. It is a totally imaginary quadratic extension of the totally real field $\mathbb Q (\zeta_n +\zeta_n^).$ The latter is the fixed field of complex conjugation, and $\mathbb Q (\zeta_n)$ is obtained from it by adjoining a square root of $\zeta_n^2+\zeta_n^-2 = (\zeta_n - \zeta_n^)^2.$
*The union Q^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 Q^R. The absolute Galois group Gal(/Q^R) is generated (as a closed subgroup) by all elements of order 2 in Gal(/Q), and Gal(/Q^CM) is a subgroup of index 2. The Galois group Gal(Q^CM/Q) has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group Gal(Q^R/Q).
*If ''V'' is a complex abelian variety of dimension ''n'', then any abelian algebra ''F'' of endomorphisms of ''V'' has rank at most 2''n'' over Z. If it has rank 2''n'' 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

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* {{cite book, first=Lawrence C., last=Washington, title=Introduction to Cyclotomic fields, publisher= Springer-Verlag, location=New York, year=1996, edition=2nd, isbn=0-387-94762-0, zbl=0966.11047

Field (mathematics)
Algebraic number theory