Editing Conjugate element (field theory)

{{About|the conjugation between the roots of a polynomial|other uses|Conjugation (disambiguation){{!}}Conjugation}}
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In [[mathematics]], in particular [[field theory (mathematics)|field theory]], the '''conjugate elements''' or '''algebraic conjugates''' of an [[algebraic element]]&nbsp;{{math|''α''}}, over a [[field extension]] {{math|''L''/''K''}}, are the [[zero of a function|root]]s of the [[minimal polynomial (field theory)|minimal polynomial]] {{math|''p''<sub>''K'',''α''</sub>(''x'')}} of {{math|''α''}} over {{math|''K''}}. Conjugate elements are commonly called '''conjugates''' in contexts where this is not ambiguous.  Normally {{math|''α''}} itself is included in the set of conjugates of&nbsp;{{math|''α''}}.

Equivalently, the conjugates of {{math|''α''}} are the images of {{math|''α''}} under the [[field automorphism]]s of {{mvar|L}} that leave fixed the elements of {{mvar|K}}. The equivalence of the two definitions is one of the starting points of [[Galois theory]].

The concept generalizes the [[complex conjugation]], since the algebraic conjugates over <math>\R</math> of a [[complex number]] are the number itself and its ''complex conjugate''.

==Example==
The cube roots of the number [[one (number)|one]] are:

: <math>\sqrt[3]{1} = \begin{cases}1 \\[3pt] -\frac{1}{2}+\frac{\sqrt{3}}{2}i \\[5pt] -\frac{1}{2}-\frac{\sqrt{3}}{2}i \end{cases} </math>

The latter two roots are conjugate elements in  {{math|'''Q'''[''i''{{sqrt|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''. 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&nbsp;''α''. If ''L'' is any [[normal extension]] of ''K'' containing&nbsp;''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 (mathematics)#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 [[Leopold Kronecker|Kronecker]] states that if ''α'' is a nonzero [[algebraic integer]] such that ''α'' and all of its conjugates in the [[complex number]]s have [[absolute value]] at most 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}}

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[[Category:Field (mathematics)]]