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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 qm
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 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
vTransitivity of trace. Prove that
TrFqrm Fq alpha TrFqm Fq TrFqrm Fqm alpha
for any alpha in Fqrm
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