Editing Field trace (section)

==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''.