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| {{More footnotes|date=May 2012}}
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| {{Technical|date=May 2012}}
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| :''For other uses, see [[Trace]]''
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| In [[mathematics]], the '''field trace''' is a [[function (mathematics)|function]] defined with respect to a [[finite extension|finite]] [[field extension]] ''L''/''K''. It is a [[linear map|''K''-linear map]] from ''L'' to ''K''. As an example, if ''L''/''K'' is a [[Galois extension]] and α is in ''L'', then the trace of α is the sum of all the [[Galois conjugate]]s of α, i.e.
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| :<math>\operatorname{Tr}_{L/K}(\alpha)=\sum_{g\in\operatorname{Gal}(L/K)}g(\alpha),</math>
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| where Gal(''L''/''K'') denotes the [[Galois group]] of ''L''/''K''. When ''K'' is the prime subfield of ''L'', the trace is called the ''absolute trace'' and otherwise it is a ''relative trace''.<ref>{{harvnb|Lidl|Niederreiter|1983|loc=p. 54}}</ref> | |
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| For a general finite extension ''L''/''K'', the trace of an element α can be defined as the [[Trace (linear algebra)|trace]] of the ''K''-linear map "multiplication by α", that is, the map from ''L'' to itself sending ''x'' to αx. If ''L''/''K'' is [[inseparable]], then the trace map is identically 0.
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| When ''L''/''K'' is separable, a formula similar to the Galois case above can be obtained. If σ{{sub|1}}, ..., σ{{sub|n}} are the distinct ''K''-linear field embeddings of ''L'' into an [[algebraically closed field]] {{overline|''F''}} containing ''K'' (where ''n'' is the degree of the extension ''L''/''K''), then
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| :<math>\operatorname{Tr}_{L/K}(\alpha)=\sum_{j=1}^n\sigma_j(\alpha).</math>
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| ==Properties of the trace==
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| As mentioned above, the trace {{nowrap|Tr{{sub|''L''/''K''}} : ''L'' → ''K''}} is a ''K''-linear map. Additionally, it behaves well in [[tower of fields|towers of fields]]: if ''M'' is a finite extension of ''L'', then the trace from ''M'' to ''K'' is just the composition of the trace from ''M'' to ''L'' with the trace from ''L'' to ''K'', i.e.
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| :<math>\operatorname{Tr}_{M/K}=\operatorname{Tr}_{L/K}\circ\operatorname{Tr}_{M/L}.</math> | |
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| ==Trace form==
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| When ''L''/''K'' is separable, the trace provides a [[duality theory]] via the '''trace form''': the map from {{nowrap|''L'' × ''L''}} to ''K'' sending (''x'', ''y'') to Tr{{sub|''L''/''K''}}(xy) is a [[nondegenerate form|nondegenerate]], [[symmetric form|symmetric]], [[bilinear form]] called the trace form. An example of where this is used is in [[algebraic number theory]] in the theory of the [[different ideal]].
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| The trace form for a finite degree field extension ''L''/''K'' has non-negative [[Signature (quadratic form)|signature]] for any [[field ordering]] of ''K''.<ref name=L38/> The converse, that every [[Witt equivalence]] class with non-negative signature contains a trace form, is true for algebraic number fields ''K''.<ref name=L38>Lorenz (2008) p.38</ref>
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| ==See also==
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| * [[Field norm]]
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| * [[Reduced trace]]
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| ==References==
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| {{reflist}}
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| * {{cite book | first1=P.E. | last1=Conner | first2=R. | last2=Perlis | title=A Survey of Trace Forms of Algebraic Number Fields | series=Series in Pure Mathematics | volume=2 | publisher=World Scientific | year=1984 | isbn=9971-966-05-0 | zbl=0551.10017 }}
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| * Section VI.5 of {{Lang Algebra|edition=3r}}
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| * {{citation|first1=Rudolf|last1=Lidl|first2=Harald|last2=Niederreiter|title=Finite Fields|series=Encyclopedia of Mathematics and its Applications|volume=20|year=1983|publisher=Addison-Wesley|place=Reading, MA|isbn=0-201-13519-1}}
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| * {{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=Springer | isbn=978-0-387-72487-4 | zbl=1130.12001 }}
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| * {{Cite book | last=Roman | first=Steven | title=Field theory | edition=Second | year=2006 | publisher=Springer | series=Graduate Texts in Mathematics | volume=158 | at=Chapter 8 | isbn=978-0-387-27677-9 | zbl=1172.12001 }}
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| {{DEFAULTSORT:Field Trace}}
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| [[Category:Field theory]] | |
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