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 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,

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 nth 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 .

[edit] References

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