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{{redirect|Conjugate elements|conjugate group elements|Conjugacy class}}
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{{refimprove|date=December 2010}}
In [[mathematics]], in particular [[field theory (mathematics)|field theory]], the '''conjugate elements''' of an [[algebraic element]] α, over a [[field extension]] ''L/K'', are the (other) roots of the [[minimal polynomial (field theory)|minimal polynomial]]
 
:''p''<sub>''K'',α</sub>(''x'')
 
of α over ''K''.
 
==Example==
The cube roots of the number [[one (number)|one]] are:
 
:<math>\sqrt[3]{1} = \begin{cases} \ \ 1 \\[8pt] -\frac{1}{2}+\frac{\sqrt{3}}{2}i \\[8pt] -\frac{1}{2}-\frac{\sqrt{3}}{2}i \end{cases} </math>
 
The latter two roots are conjugate elements in ''L''/''K'' = '''Q'''[√3, ''i'']/'''Q'''[√3] with minimal polynomial
 
: <math> \left(x+\frac{1}{2}\right)^2+\frac{3}{4}=x^2+x+1.</math>
 
==Properties==
If ''K'' is given inside an [[algebraically closed field]] ''C'', then the conjugates can be taken inside ''C''. Usually one includes α itself in the set of conjugates. If no such ''C'' is specified, one can take the conjugates in some relatively small field ''L''. The smallest possible choice for ''L'' is to take a [[splitting field]] over ''K'' of ''p''<sub>''K'',α</sub>, containing α. If ''L'' is any [[normal extension]] of ''K'' containing α, then by definition it already contains such a splitting field.
 
Given then a normal extension ''L'' of ''K'', with [[Galois group|automorphism group]] Aut(''L''/''K'') = ''G'', and containing α, any element ''g''(α) for ''g'' in ''G'' will be a conjugate of α, since the [[automorphism]] ''g'' sends roots of ''p'' to roots of ''p''. Conversely any conjugate β of α is of this form: in other words, ''G'' acts [[Group_action#Types_of_actions|transitively]] on the conjugates. This follows as ''K''(α) is ''K''-isomorphic to ''K''(β) by irreducibility of the minimal polynomial, and any isomorphism of fields ''F'' and ''F{{'}}'' that maps polynomial ''p'' to ''p{{'}}'' can be extended to an isomorphism of the splitting fields of ''p'' over ''F'' and ''p{{'}}'' over ''F{{'}}'', respectively.
 
In summary, the conjugate elements of α are found, in any normal extension ''L'' of ''K'' that contains ''K''(α), as the set of elements ''g''(α) for ''g'' in Aut(''L''/''K''). The number of repeats in that list of each element is the separable degree [''L'':''K''(α)]<sub>sep</sub>.
 
A theorem of [[Kronecker]] states that if α is an [[algebraic integer]] such that α and all of its conjugates in the [[complex number]]s have [[absolute value]] 1, then α is a [[root of unity]]. There are quantitative forms of this, stating more precisely bounds (depending on degree) on the largest absolute value of a conjugate that imply that an algebraic integer is a  root of unity.
 
==References==
*David S. Dummit, Richard M. Foote, ''Abstract algebra'', 3rd ed., Wiley, 2004.
 
==External links==
* {{MathWorld |title=Conjugate Elements |id=ConjugateElements}}
 
{{DEFAULTSORT:Conjugate Element (Field Theory)}}
[[Category:Field theory]]

Latest revision as of 15:12, 27 December 2014

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