▁cup - product ▁of ▁the ▁first ▁Gal ois ▁coh om ology ▁groups . ▁ ▁For ▁various ▁approaches ▁to ▁local ▁class ▁field ▁theory ▁see ▁Ch . ▁IV ▁and ▁sect . ▁ 7 ▁Ch . ▁IV ▁of ▁▁ ▁They ▁include ▁the ▁H asse ▁approach ▁of ▁using ▁the ▁Bra uer ▁group , ▁coh om ological ▁approaches , ▁the ▁explicit ▁methods ▁of ▁J ürgen ▁Ne uk ir ch , ▁Mich iel ▁Haz ew inkel , ▁the ▁Lub in - T ate ▁theory ▁and ▁others . ▁ ▁General izations ▁of ▁local ▁class ▁field ▁theory ▁General izations ▁of ▁local ▁class ▁field ▁theory ▁to ▁local ▁fields ▁with ▁quasi - finite ▁resid ue ▁field ▁were ▁easy ▁extensions ▁of ▁the ▁theory , ▁obtained ▁by ▁G . ▁Wh ap les ▁in ▁the ▁ 1 9 5 0 s , ▁see ▁chapter ▁V ▁of . ▁ ▁Exp licit ▁p - class ▁field ▁theory ▁for ▁local ▁fields ▁with ▁perfect ▁and ▁imper fect ▁resid ue ▁fields ▁which ▁are ▁not ▁finite ▁has ▁to ▁deal ▁with ▁the ▁new ▁issue ▁of ▁norm ▁groups ▁of ▁infinite ▁index . ▁App ropri ate ▁theories ▁were ▁constructed ▁by ▁Ivan ▁F esen ko . ▁F esen ko ' s ▁non comm ut ative ▁local ▁class ▁field ▁theory ▁for ▁ar ithmet ically ▁prof in ite ▁Gal ois ▁extensions ▁of ▁local ▁fields ▁studies ▁appropriate ▁local ▁reci pro city ▁c oc ycle ▁map ▁and ▁its ▁properties . ▁This ▁arithmetic ▁theory ▁can ▁be ▁viewed ▁as ▁an ▁alternative ▁to ▁the ▁representation ▁theoretical ▁local ▁Lang lands ▁correspond ence . ▁ ▁High er ▁local ▁class ▁field ▁theory ▁For ▁a ▁higher -