Editing Field trace

{{other uses|Trace (disambiguation)}}
In [[mathematics]], the '''field trace''' is a particular [[function (mathematics)|function]] defined with respect to a [[finite extension|finite]] [[field extension]] ''L''/''K'', which is a [[linear map|''K''-linear map]] from ''L'' onto ''K''.

==Definition==
Let ''K'' be a [[field (mathematics)|field]] and ''L'' a finite extension (and hence an [[algebraic extension]]) of ''K''. ''L'' can be viewed as a [[vector space]] over ''K''. Multiplication by ''α'', an element of ''L'',
:<math>m_\alpha:L\to L \text{ given by } m_\alpha (x) = \alpha x</math>,
is a ''K''-[[linear transformation]] of this vector space into itself. The ''trace'', '''Tr'''<sub>''L''/''K''</sub>(''α''), is defined as the [[Trace (linear algebra)|trace]] (in the [[linear algebra]] sense) of this linear transformation.<ref name=ROT940>{{harvnb|Rotman|2002|loc=p. 940}}</ref>

For ''α'' in ''L'', let ''σ''{{sub|1}}(''α''), ..., ''σ''{{sub|''n''}}(''α'') be the [[root of a polynomial|roots]] (counted with multiplicity) of the [[minimal polynomial (field theory)|minimal polynomial]] of ''α'' over ''K''  (in some extension field of ''K''). Then
:<math>\operatorname{Tr}_{L/K}(\alpha) = [L:K(\alpha)]\sum_{j=1}^n\sigma_j(\alpha).</math>
If ''L''/''K'' is [[separable extension|separable]] then each root appears only once<ref>{{harvnb|Rotman|2002|loc=p. 941}}</ref> (however this does not mean the coefficient above is one; for example if ''α'' is the identity element 1 of ''K'' then the trace is [''L'':''K''] times 1).

More particularly, 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'',<ref name="ROT940" /> i.e.,

:<math>\operatorname{Tr}_{L/K}(\alpha)=\sum_{\sigma\in\operatorname{Gal}(L/K)}\sigma(\alpha),</math>
where Gal(''L''/''K'') denotes the [[Galois group]] of ''L''/''K''.

==Example==
Let <math>L = \mathbb{Q}(\sqrt{d})</math> be a [[quadratic extension]] of <math>\mathbb{Q}</math>. Then a [[basis (linear algebra)|basis]] of <math>L/\mathbb{Q}</math> is <math>\{1, \sqrt{d}\}.</math> If <math>\alpha = a + b\sqrt{d}</math> then the [[matrix (mathematics)|matrix]] of <math>m_{\alpha}</math> is:
:<math>\left [ \begin{matrix} a & bd \\ b & a \end{matrix} \right ]</math>,
and so, <math>\operatorname{Tr}_{L/\mathbb{Q}}(\alpha) = [L:\mathbb{Q}(\alpha)]\left( \sigma_1(\alpha) + \sigma_2(\alpha)\right)
 = 1\times \left( \sigma_1(\alpha) + \overline{\sigma_1}(\alpha)\right)
 = a+b\sqrt{d} + a-b\sqrt{d} = 2a</math>.<ref name=ROT940/> The minimal polynomial of ''α'' is {{nowrap|''X''{{i sup|2}} − 2''a''&thinsp;''X'' + (''a''<sup>2</sup> − ''db''<sup>2</sup>)}}.

==Properties of the trace==
Several properties of the trace function hold for any finite extension.<ref>{{harvnb|Roman|2006|p=151}}</ref>

The trace {{nowrap|Tr{{sub|''L''/''K''}} : ''L'' → ''K''}} is a ''K''-[[linear map]] (a ''K''-linear functional), that is
:<math>\operatorname{Tr}_{L/K}(\alpha a + \beta b) = \alpha \operatorname{Tr}_{L/K}(a)+ \beta \operatorname{Tr}_{L/K}(b) \text{ for all }\alpha, \beta \in K</math>.

If {{nowrap|''α'' ∈ ''K''}} then <math>\operatorname{Tr}_{L/K}(\alpha) = [L:K] \alpha.</math>

Additionally, trace 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 [[function composition|composition]] of the trace from ''M'' to ''L'' with the trace from ''L'' to ''K'', i.e.
:<math>\operatorname{Tr}_{M/K}=\operatorname{Tr}_{L/K}\circ\operatorname{Tr}_{M/L}</math>.

==Finite fields==
Let ''L'' = GF(''q''<sup>''n''</sup>) be a finite extension of a [[finite field]] ''K'' = GF(''q''). Since ''L''/''K'' is a [[Galois extension]], if ''α'' is in ''L'', then the trace of ''α'' is the sum of all the [[Galois conjugate]]s of ''α'', i.e.<ref name=LN54>{{harvnb|Lidl|Niederreiter|1997|loc=p.54}}</ref>

:<math>\operatorname{Tr}_{L/K}(\alpha)=\alpha + \alpha^q + \cdots + \alpha^{q^{n-1}}.</math>

In this setting we have the additional properties:<ref>{{harvnb|Mullen|Panario|2013|loc=p. 21}}</ref>
* <math>\operatorname{Tr}_{L/K}(a^q) = \operatorname{Tr}_{L/K}(a) \text{ for } a \in L</math>.
* For any <math>\alpha \in K</math>, there are exactly <math> q^{n-1}</math> elements <math>b\in L</math> with <math>\operatorname{Tr}_{L/K}(b) = \alpha</math>. 

''Theorem''.<ref name=LN56>{{harvnb|Lidl|Niederreiter|1997|loc=p.56}}</ref> For ''b'' ∈ ''L'', let ''F''<sub>''b''</sub> be the map <math>a \mapsto \operatorname{Tr}_{L/K}(ba).</math> Then {{nowrap|''F''<sub>''b''</sub> ≠ ''F''<sub>''c''</sub>}} if {{nowrap|''b'' ≠ ''c''}}. Moreover, the ''K''-linear transformations from ''L'' to ''K'' are exactly the maps of the form ''F''<sub>''b''</sub> as ''b'' varies over the field ''L''.

When ''K'' is the [[prime subfield]] of ''L'', the trace is called the ''absolute trace'' and otherwise it is a ''relative trace''.<ref name=LN54/>

===Application===
A [[quadratic equation]], {{nowrap|1=''ax''{{i sup|2}} + ''bx'' + ''c'' = 0}} with ''a''&nbsp;≠&nbsp;0, and coefficients in the finite field <math>\operatorname{GF}(q) = \mathbb{F}_q</math> has either 0, 1 or 2 roots in GF(''q'') (and two roots, counted with multiplicity, in the quadratic extension GF(''q''<sup>2</sup>)). If the [[characteristic (algebra)|characteristic]] of GF(''q'') is [[parity (mathematics)|odd]], the [[discriminant]] {{nowrap|1=Δ = ''b''<sup>2</sup> − 4''ac''}} indicates the number of roots in GF(''q'') and the classical [[quadratic formula]] gives the roots. However, when GF(''q'') has [[parity (mathematics)|even]] characteristic (i.e., {{nowrap|1=''q'' = 2<sup>''h''</sup>}} for some positive [[integer]] ''h''), these formulas are no longer applicable.

Consider the quadratic equation {{nowrap|1=''ax''{{i sup|2}} + ''bx'' + c = 0}} with coefficients in the finite field GF(2<sup>''h''</sup>).<ref>{{harvnb|Hirschfeld|1979|loc=pp. 3-4}}</ref> If ''b'' = 0 then this equation has the unique solution <math>x = \sqrt{\frac{c}{a}}</math> in GF(''q''). If {{nowrap|''b'' ≠ 0}} then the substitution {{nowrap|1=''y'' = ''ax''/''b''}} converts the quadratic equation to the form:
:<math>y^2 + y + \delta = 0, \text { where } \delta = \frac{ac}{b^2}.</math>
This equation has two solutions in GF(''q'') [[if and only if]] the absolute trace <math>\operatorname{Tr}_{GF(q)/GF(2)}(\delta) = 0.</math> In this case, if ''y''&nbsp;=&nbsp;''s'' is one of the solutions, then ''y''&nbsp;=&nbsp;''s''&nbsp;+&thinsp;1 is the other. Let ''k'' be any element of GF(''q'') with <math>\operatorname{Tr}_{GF(q)/GF(2)}(k) = 1.</math> Then a solution to the equation is given by:
:<math> y = s = k \delta^2 + (k + k^2)\delta^4 + \ldots + (k + k^2 + \ldots + k^{2^{h-2}})\delta^{2^{h-1}}.</math>
When ''h'' = 2''m'''&nbsp;+&thinsp;1, a solution is given by the simpler expression:
:<math> y = s = \delta + \delta^{2^2} + \delta^{2^4} + \ldots + \delta^{2^{2m}}.</math>

==Trace form==
When ''L''/''K'' is separable, the trace provides a [[duality theory]] via the '''trace form''': the map from {{nowrap|''L'' × ''L''}} to ''K'' sending {{nowrap|(''x'', ''y'')}} to Tr{{sub|''L''/''K''}}(''xy'') is a [[nondegenerate form|nondegenerate]], [[symmetric bilinear form]] called the trace form. If ''L''/''K'' is a Galois extension, the trace form is invariant with respect to the Galois group.

The trace form is used in [[algebraic number theory]] in the theory of the [[different ideal]].

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 (logic)|converse]], that every [[Witt ring (forms)|Witt equivalence]] class with non-negative signature contains a trace form, is true for [[algebraic number field]]s ''K''.<ref name=L38>Lorenz (2008) p.38</ref>

If ''L''/''K'' is an [[inseparable extension]], then the trace form is identically 0.<ref>{{harvnb|Isaacs|1994|loc=p. 369}} as footnoted in {{harvnb|Rotman|2002|loc=p. 943}}</ref>

==See also==
* [[Field norm]]
* [[Reduced trace]]

==Notes==
{{reflist|3}}

==References==
* {{citation|first=J.W.P.|last=Hirschfeld|year=1979|title=Projective Geometries over Finite Fields|series=Oxford Mathematical Monographs|publisher=Oxford University Press|isbn=0-19-853526-0|url-access=registration|url=https://archive.org/details/projectivegeomet0000hirs}}
* {{citation|first=I.M.|last=Isaacs|title=Algebra, A Graduate Course|year=1994|publisher=Brooks/Cole Publishing}}
* {{citation | first1=Rudolf | last1=Lidl | first2=Harald | last2=Niederreiter | author2-link=Harald Niederreiter | title=Finite Fields | series=Encyclopedia of Mathematics and its Applications | volume=20 | year=1997 | origyear=1983 | edition=Second | publisher=[[Cambridge University Press]] | isbn=0-521-39231-4 | zbl=0866.11069 | url-access=registration | url=https://archive.org/details/finitefields0000lidl_a8r3 }}
* {{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 }} 
* {{citation|first1=Gary L.|last1=Mullen|first2=Daniel|last2=Panario|title=Handbook of Finite Fields|year=2013|publisher=CRC Press|isbn=978-1-4398-7378-6}}
* {{citation | 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 }}
* {{citation|first=Joseph J.|last=Rotman|title=Advanced Modern Algebra|year=2002|publisher=Prentice Hall|isbn=978-0-13-087868-7}}

==Further reading==
* {{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 }}
* Section VI.5 of {{Lang Algebra|edition=3r}}

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[[Category:Field (mathematics)]]