CM-field

In mathematics, a CM-field is a particular type of number field "K", so named for a close connection to the theory of complex multiplication. Another name used is "J-field". Specifically, "K" is a totally imaginary quadratic extension of a totally real field number field. In other words, there is a subfield "K"* of "K", such that "K" is generated over "K"* by a single square root of an element, say

:β = $sqrt\{alpha\}$

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 "K"* into the real number field,

:&sigma;*(α) < 0.

One feature of a CM-field is that complex conjugation on $mathbb\; C$ induces an automorphism on the field which is independent of the embedding into $mathbb\; C$. In the notation given, it must change the sign of β.

One of the most important example 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

:$mathbb\; Q\; (zeta\_n\; +zeta\_n^\{-1\})$.

This is the fixed field of complex conjugation, and we obtain $mathbb\; Q\; (zeta\_n)$ back from it by adjoining a square root of

:$zeta\_n^2+zeta\_n^\{-2\}-2$.

References

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