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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 [[quadratic extension]] of a [[totally real field]] which is [[totally imaginary field|totally imaginary]], i.e. for which there is no embedding of ''K'' into <math>\mathbb R </math>.
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| In other words, there is a subfield <math>K'</math> of ''K'' such that ''K'' is generated over <math>K'</math> by a single square root of an element, say
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| β = <math>\sqrt{\alpha} </math>,
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| in such a way that the [[minimal polynomial (field theory)|minimal polynomial]] of β over the [[rational number field]] <math> \mathbb Q</math> has all its roots non-real complex numbers. For this α should be chosen ''totally negative'', so that for each embedding σ of <math>K'</math> into the real number field,
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| σ(α) < 0.
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| ==Properties==
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| One feature of a CM-field is that [[complex conjugation]] on <math>\mathbb C </math> induces an automorphism on the field which is independent of the embedding into <math>\mathbb C</math>. In the notation given, it must change the sign of β. | |
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| A number field ''F'' is a CM-field if and only if it has a "units defect", i.e. if it contains a proper subfield <math>F'</math> whose unit group has the same <math>\mathbb Z</math>-rank as that of ''F''.<ref>{{cite journal | last=Remak | first=Robert | title=Über algebraische Zahlkörper mit schwachem Einheitsdefekt | language=German | journal=Compositio Math. | volume=12 | year=1954 | pages=35–80 | zbl=0055.26805 }}</ref> (In fact, <math>F'</math> is the totally real subfield of ''F'' mentioned above.) This follows from [[Dirichlet's unit theorem]].
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| ==Examples==
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| * 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
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| * One of the most important examples of a CM-field is the [[cyclotomic field]] <math> \mathbb Q (\zeta_n) </math>, which is generated by a primitive n<sup>th</sup> [[root of unity]]. It is a totally imaginary [[quadratic extension]] of the [[totally real field]] <math> \mathbb Q (\zeta_n +\zeta_n^{-1}). </math> The latter is the fixed field of [[complex conjugation]], and <math> \mathbb Q (\zeta_n) </math> is obtained from it by adjoining a square root of <math> \zeta_n^2+\zeta_n^{-2}-2 = (\zeta_n - \zeta_n^{-1})^2. </math>
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| ==References==
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| {{reflist}}
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| * {{cite book|first=Lawrence C.|last=Washington|title=Introduction to Cyclotomic fields|publisher=Springer-Verlag|location=New York|year=1996|edition=2nd edition|isbn=0-387-94762-0|zbl=0966.11047}}
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| [[Category:Field theory]]
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| [[Category:Algebraic number theory]]
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