Define TrFqm $/$Fq $(\backslash$alpha $)=\backslash$alpha $+\backslash$alpha q $++\backslash$alpha qm$1$

for any $\backslash$alpha in Fqm $.$ The element

TrFqm $/$Fq $(\backslash$alpha $)$ is called the trace of $\backslash$alpha with respect to the extension Fqm $/$Fq $.$

$($i$)$ Show that TrFqm $/$Fq $(\backslash$alpha $)$ is an element of Fq for all $\backslash$alpha in Fqm $.$

$($ii$)$ Show that the map

TrFqm $/$Fq : Fqm $->$ Fq $,\backslash$alpha $->$ TrFqm $/$Fq $(\backslash$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 $\backslash$beta in Fqm $.$ Prove that TrFqm $/$Fq $(\backslash$beta $)=0$ if and only if there exists

an element $\backslash$gamma in Fqm such that $\backslash$beta $=\backslash$gamma q $\backslash$gamma $.($Note: this statement

is commonly referred to as the additive form of Hilbert$$s Theorem

$90.)$

$($v$)($Transitivity of trace.$)$ Prove that

TrFqrm $/$Fq $(\backslash$alpha $)=$ TrFqm $/$Fq $($TrFqrm $/$Fqm $(\backslash$alpha $))$

for any $\backslash$alpha in Fqrm $.$