CM-field

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In mathematics, a CM-field is a particular type of number field K, so named for a close connection ton 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,

σ*(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's a totally imaginary quadratic extension of totally real field $\mathbb Q (\zeta_n +\zeta_n^{-1})$ . Note that it's the fixed field of complex conjugation and we obtain $\mathbb Q (\zeta_n)$ back by adjoining a square root of $\zeta_n^2+\zeta_n^{-2}-2$ .