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Define TrFqm / Fq ( alpha ) = alpha + alpha q + + alpha qm 1 for any alpha

Define TrFqm /Fq (\alpha )=\alpha +\alpha q ++\alpha qm1
for any \alpha in Fqm . The element
TrFqm /Fq (\alpha ) is called the trace of \alpha with respect to the extension Fqm /Fq .
(i) Show that TrFqm /Fq (\alpha ) is an element of Fq for all \alpha in Fqm .
(ii) Show that the map
TrFqm /Fq : Fqm -> Fq ,\alpha -> TrFqm /Fq (\alpha )
is an Fq -linear transformation, where both Fqm and Fq are viewed as
vector spaces over Fq .
(iii) Show that TrFqm /Fq is surjective.
(iv) Let \beta in Fqm . Prove that TrFqm /Fq (\beta )=0 if and only if there exists
an element \gamma in Fqm such that \beta =\gamma q \gamma .(Note: this statement
is commonly referred to as the additive form of Hilberts Theorem
90.)
(v)(Transitivity of trace.) Prove that
TrFqrm /Fq (\alpha )= TrFqm /Fq (TrFqrm /Fqm (\alpha ))
for any \alpha in Fqrm .

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