▁ ▁over ▁K ▁is ▁equivalent ▁to ▁a ▁direct ▁sum ▁ ▁of ▁n ons ing ular ▁binary ▁forms . ▁This ▁was ▁shown ▁by ▁Ar f , ▁but ▁it ▁had ▁been ▁earlier ▁observed ▁by ▁Dick son ▁in ▁the ▁case ▁of ▁finite ▁fields ▁of ▁characteristic ▁ 2 . ▁The ▁Ar f ▁invariant ▁Ar f () ▁is ▁defined ▁to ▁be ▁the ▁sum ▁of ▁the ▁Ar f ▁invari ants ▁of ▁the ▁. ▁By ▁definition , ▁this ▁is ▁a ▁cos et ▁of ▁K ▁mod ulo ▁U . ▁Ar f ▁showed ▁that ▁indeed ▁Ar f () ▁does ▁not ▁change ▁if ▁ ▁is ▁replaced ▁by ▁an ▁equivalent ▁quadratic ▁form , ▁which ▁is ▁to ▁say ▁that ▁it ▁is ▁an ▁invariant ▁of ▁. ▁ ▁The ▁Ar f ▁invariant ▁is ▁add itive ; ▁in ▁other ▁words , ▁the ▁Ar f ▁invariant ▁of ▁an ▁orthogonal ▁sum ▁of ▁two ▁quadratic ▁forms ▁is ▁the ▁sum ▁of ▁their ▁Ar f ▁invari ants . ▁ ▁For ▁a ▁field ▁K ▁of ▁characteristic ▁ 2 , ▁Art in - Sch re ier ▁theory ▁ident ifies ▁the ▁quot ient ▁group ▁of ▁K ▁by ▁the ▁subgroup ▁U ▁above ▁with ▁the ▁Gal ois ▁coh om ology ▁group ▁H 1 ( K , ▁F 2 ). ▁In ▁other ▁words , ▁the ▁non zero ▁elements ▁of ▁K / U ▁are ▁in ▁one - to - one ▁correspond ence ▁with ▁the ▁separ able ▁quadratic ▁extension ▁fields ▁of ▁K . ▁So ▁the ▁Ar f ▁invariant ▁of ▁a ▁n ons ing ular ▁quadratic ▁form ▁over ▁K ▁is ▁either ▁zero ▁or ▁it ▁describes ▁a ▁separ able ▁quadratic ▁extension ▁field ▁of