▁K . ▁This ▁is ▁analog ous ▁to ▁the ▁disc rimin ant ▁of ▁a ▁n ons ing ular ▁quadratic ▁form ▁over ▁a ▁field ▁F ▁of ▁characteristic ▁not ▁ 2 . ▁In ▁that ▁case , ▁the ▁disc rimin ant ▁takes ▁values ▁in ▁F */ ( F *) 2 , ▁which ▁can ▁be ▁identified ▁with ▁H 1 ( F , ▁F 2 ) ▁by ▁K um mer ▁theory . ▁ ▁Ar f ' s ▁main ▁results ▁ ▁If ▁the ▁field ▁K ▁is ▁perfect , ▁then ▁every ▁n ons ing ular ▁quadratic ▁form ▁over ▁K ▁is ▁uniqu ely ▁determined ▁( up ▁to ▁equivalence ) ▁by ▁its ▁dimension ▁and ▁its ▁Ar f ▁invariant . ▁In ▁particular , ▁this ▁holds ▁over ▁the ▁field ▁F 2 . ▁In ▁this ▁case , ▁the ▁subgroup ▁U ▁above ▁is ▁zero , ▁and ▁hence ▁the ▁Ar f ▁invariant ▁is ▁an ▁element ▁of ▁the ▁base ▁field ▁F 2 ; ▁it ▁is ▁either ▁ 0 ▁or ▁ 1 . ▁ ▁If ▁the ▁field ▁K ▁of ▁characteristic ▁ 2 ▁is ▁not ▁perfect ▁( that ▁is , ▁K ▁is ▁different ▁from ▁its ▁sub field ▁K 2 ▁of ▁squares ), ▁then ▁the ▁Cl iff ord ▁algebra ▁is ▁another ▁important ▁invariant ▁of ▁a ▁quadratic ▁form . ▁A ▁corrected ▁version ▁of ▁Ar f ' s ▁original ▁statement ▁is ▁that ▁if ▁the ▁degree ▁[ K : ▁K 2 ] ▁is ▁at ▁most ▁ 2 , ▁then ▁every ▁quadratic ▁form ▁over ▁K ▁is ▁completely ▁character ized ▁by ▁its ▁dimension , ▁its ▁Ar f ▁invariant ▁and ▁its ▁Cl iff ord ▁algebra . ▁Ex amples