▁of ▁ Γ v ▁in ▁ Γ w , ▁e ( w / v )   =   [ Γ w   :   Γ v ], ▁is ▁called ▁the ▁reduced ▁ram ification ▁index ▁of ▁w ▁over ▁v . ▁It ▁satisfies ▁e ( w / v )   ≤   [ L   :   K ] ▁( the ▁degree ▁of ▁the ▁extension ▁L / K ). ▁The ▁relative ▁degree ▁of ▁w ▁over ▁v ▁is ▁defined ▁to ▁be ▁f ( w / v )   =   [ R w / m w   :   R v / mv ] ▁( the ▁degree ▁of ▁the ▁extension ▁of ▁resid ue ▁fields ). ▁It ▁is ▁also ▁less ▁than ▁or ▁equal ▁to ▁the ▁degree ▁of ▁L / K . ▁When ▁L / K ▁is ▁separ able , ▁the ▁ram ification ▁index ▁of ▁w ▁over ▁v ▁is ▁defined ▁to ▁be ▁e ( w / v ) pi , ▁where ▁pi ▁is ▁the ▁in separ able ▁degree ▁of ▁the ▁extension ▁R w / m w ▁over ▁R v / mv . ▁ ▁Complete ▁val ued ▁fields ▁ ▁When ▁the ▁ordered ▁ab elian ▁group ▁ ▁is ▁the ▁add itive ▁group ▁of ▁the ▁integers , ▁the ▁associated ▁valu ation ▁is ▁equivalent ▁to ▁an ▁absolute ▁value , ▁and ▁hence ▁indu ces ▁a ▁metric ▁on ▁the ▁field ▁. ▁If ▁ ▁is ▁complete ▁with ▁respect ▁to ▁this ▁metric , ▁then ▁it ▁is ▁called ▁a ▁complete ▁val ued ▁field . ▁If ▁K ▁is ▁not ▁complete , ▁one ▁can ▁use ▁the ▁valu ation ▁to