▁with ▁an ▁even ▁number ▁of ▁terms ▁is ▁divis ible ▁by ▁ ▁( it ▁has ▁ 1 ▁as ▁a ▁root ). ▁ ▁An ▁ir reducible ▁polynomial ▁F ( x ) ▁of ▁degree ▁m ▁over ▁G F ( p ), ▁where ▁p ▁is ▁prime , ▁is ▁a ▁primitive ▁polynomial ▁if ▁the ▁smallest ▁positive ▁integer ▁n ▁such ▁that ▁F ( x ) ▁div ides ▁ ▁is ▁. ▁ ▁Over ▁G F ( pm ) ▁there ▁are ▁exactly ▁ ▁primitive ▁polynomials ▁of ▁degree ▁m , ▁where ▁ φ ▁is ▁E uler ' s ▁tot ient ▁function . ▁ ▁A ▁primitive ▁polynomial ▁of ▁degree ▁m ▁has ▁m ▁different ▁roots ▁in ▁G F ( pm ), ▁which ▁all ▁have ▁order ▁. ▁This ▁means ▁that , ▁if ▁α ▁is ▁such ▁a ▁root , ▁then ▁ ▁and ▁ ▁for ▁. ▁ ▁The ▁primitive ▁polynomial ▁F ( x ) ▁of ▁degree ▁m ▁of ▁a ▁primitive ▁element ▁α ▁in ▁G F ( pm ) ▁has ▁explicit ▁form ▁. ▁ ▁Us age ▁ ▁Field ▁element ▁representation ▁Prim itive ▁polynomials ▁are ▁used ▁in ▁the ▁representation ▁of ▁elements ▁of ▁a ▁finite ▁field . ▁ ▁If ▁α ▁in ▁G F ( pm ) ▁is ▁a ▁root ▁of ▁a ▁primitive ▁polynomial ▁F ( x ) ▁then ▁since ▁the ▁order ▁of ▁α ▁is ▁ ▁that ▁means ▁that ▁all ▁non zero ▁elements ▁of ▁G F ( pm ) ▁can ▁be ▁represented ▁as ▁success ive ▁powers ▁of ▁α : ▁ ▁When ▁these ▁elements ▁are ▁reduced ▁mod ulo ▁F ( x ), ▁they ▁provide ▁the ▁polynomial ▁basis ▁representation ▁of ▁all ▁the ▁elements