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 & Taniyama 1961).

Contents

• 
  1 Formal definition
  


• 
  2 Properties
  


• 
  3 Examples
  


• 
  4 References
  

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 C{displaystyle mathbb {C} } lies entirely within R{displaystyle mathbb {R} }, but there is no embedding of K into R{displaystyle 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 = α{displaystyle {sqrt {alpha }}},
in such a way that the minimal polynomial of β over the rational number field Q{displaystyle 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 K′{displaystyle K'} into the real number field,
σ(α) < 0.

Properties

One feature of a CM-field is that complex conjugation on C{displaystyle mathbb {C} } induces an automorphism on the field which is independent of its embedding into C{displaystyle 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 Z{displaystyle mathbb {Z} }-rank as that of K (Remak 1954). 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 Q(ζn){displaystyle 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 Q(ζn+ζn−1).{displaystyle mathbb {Q} (zeta _{n}+zeta _{n}^{-1}).} The latter is the fixed field of complex conjugation, and Q(ζn){displaystyle mathbb {Q} (zeta _{n})} is obtained from it by adjoining a square root of ζn2+ζn−2−2=(ζn−ζn−1)2.{displaystyle zeta _{n}^{2}+zeta _{n}^{-2}-2=(zeta _{n}-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(Q/Q^R) is generated (as a closed subgroup) by all elements of order 2 in Gal(Q/Q), and Gal(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.

References

• 
  Remak, Robert (1954), "Über algebraische Zahlkörper mit schwachem Einheitsdefekt", Compositio Mathematica (in German), 12: 35–80, Zbl 0055.26805.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, 11, Princeton, N.J.: Princeton University Press
  


• 
  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: The Mathematical Society of Japan, MR 0125113
  


• 
  Washington, Lawrence C. (1996). Introduction to Cyclotomic fields (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0. Zbl 0966.11047.
  

This page is only for reference, If you need detailed information, please check here