<<FiniteFields`

F16=GF[2,{1,1,0,0,1}];

Table[{i,b^i,
ElementToPolynomial[Power[F16[{0,1,0,0}],i],b]
//TraditionalForm,
BaseForm[ElementToPolynomial[Power[F16[{0,1,0,0}],i],b]/. b->2,
2],
ElementToPolynomial[Power[F16[{0,1,0,0}],i],b]/. b->2},
{i,1,15}]//TableForm

F256=GF[2,{1,0,1,1,1,0,0,0,1}];

Table[{i,g^i,ElementToPolynomial[Power[F256[{0,1,0,0}],i],g]//TraditionalForm,BaseForm[ElementToPolynomial[Power[F256[{0,1,0,0}],i],g]/. g->2,2],ElementToPolynomial[Power[F256[{0,1,0,0}],i],g]/. g->2},{i,1,255}]//TableForm