In
mathematics, a CM-field is a particular type of
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
, so named for a close connection to the theory of
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
. 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
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''K''/''F'' where the base field ''F'' is
totally real
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer poly ...
but ''K'' is
totally imaginary. I.e., every embedding of ''F'' into
lies entirely within
, but there is no embedding of ''K'' into
.
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 =
,
in such a way that the
minimal polynomial of β over the
rational number field
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
has all its roots non-real complex numbers. For this α should be chosen ''totally negative'', so that for each embedding σ of
into the real number field,
σ(α) < 0.
Properties
One feature of a CM-field is that
complex conjugation on
induces an automorphism on the field which is independent of its embedding into
. 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
-rank as that of ''K'' . In fact, ''F'' is the totally real subfield of ''K'' mentioned above. This follows from
Dirichlet's unit theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a pos ...
.
Examples
* The simplest, and motivating, example of a CM-field is an
imaginary quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...
, 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
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
, which is generated by a primitive nth
root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important ...
. It is a totally imaginary
quadratic extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of the
totally real field
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyno ...
The latter is the fixed field of
complex conjugation, and
is obtained from it by adjoining a square root of
*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
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' t ...
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
.
References
*
*
*
* {{cite book, first=Lawrence C., last=Washington, title=Introduction to Cyclotomic fields, publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=New York, year=1996, edition=2nd, isbn=0-387-94762-0, zbl=0966.11047
Field (mathematics)
Algebraic number theory