CM-field

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

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 = $\backslash sqrt\{\backslash alpha\}$,

in such a way that the minimal polynomial of β over the rational number field $\backslash mathbb\; Q$ 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.

One feature of a CM-field is that complex conjugation on $\backslash mathbb\; C$ induces an automorphism on the field which is independent of its embedding into $\backslash mathbb\; C$. 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 $\backslash mathbb\; Z$-rank as that of K {{harv|Remak|1954}}. In fact, F is the totally real subfield of K mentioned above. This follows from Dirichlet's unit theorem.

• 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 $\backslash mathbb\; Q\; (\backslash zeta\_n)$, which is generated by a primitive nth root of unity. It is a totally imaginary quadratic extension of the totally real field $\backslash mathbb\; Q\; (\backslash zeta\_n\; +\backslash zeta\_n^\{-1\}).$ The latter is the fixed field of complex conjugation, and $\backslash mathbb\; Q\; (\backslash zeta\_n)$ is obtained from it by adjoining a square root of $\backslash zeta\_n^2+\backslash zeta\_n^\{-2\}-2\; =\; (\backslash zeta\_n\; -\; \backslash zeta\_n^\{-1\})^2.$
• 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({{overline|Q}}/Q^R) is generated (as a closed subgroup) by all elements of order 2 in Gal({{overline|Q}}/Q), and Gal({{overline|Q}}/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 2n over 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 $x^4\; +\; x^3\; -\; x^2\; -\; x\; +\; 1$.

{{reflist}}

• {{citation | first=Robert | last=Remak | year=1954 | title=Über algebraische Zahlkörper mit schwachem Einheitsdefekt | language=German | journal=Compositio Mathematica | volume=12 | pages=35–80 | zbl=0055.26805 }}
• {{citation|last=Shimura|first= Goro |title=Introduction to the arithmetic theory of automorphic functions|series= Publications of the Mathematical Society of Japan|volume= 11|publisher= Princeton University Press|place= Princeton, N.J.|year= 1971}}
• {{citation|mr=0125113

|last=Shimura|first= Goro|last2= Taniyama|first2= Yutaka

|title=Complex multiplication of abelian varieties and its applications to number theory

|series=Publications of the Mathematical Society of Japan|volume= 6 |publisher=The Mathematical Society of Japan|place= Tokyo |year=1961 }}

• {{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}}

Category:Field (mathematics)

Category:Algebraic number theory

Category:Complex numbers