▁extension ▁L ▁of ▁K ▁the ▁reci pro city ▁map ▁indu ces ▁an ▁isomorphism ▁of ▁the ▁quot ient ▁group ▁K × / N ( L × ) ▁of ▁K × ▁by ▁the ▁norm ▁group ▁N ( L × ) ▁of ▁the ▁extension ▁L × ▁to ▁the ▁Gal ois ▁group ▁Gal ( L / K ) ▁of ▁the ▁extension . ▁ ▁The ▁existence ▁theorem ▁in ▁local ▁class ▁field ▁theory ▁establish es ▁a ▁one - to - one ▁correspond ence ▁between ▁open ▁sub groups ▁of ▁finite ▁index ▁in ▁the ▁multip lic ative ▁group ▁K × ▁and ▁finite ▁ab elian ▁extensions ▁of ▁the ▁field ▁K . ▁For ▁a ▁finite ▁ab elian ▁extension ▁L ▁of ▁K ▁the ▁corresponding ▁open ▁subgroup ▁of ▁finite ▁index ▁is ▁the ▁norm ▁group ▁N ( L × ). ▁The ▁reci pro city ▁map ▁sends ▁higher ▁groups ▁of ▁units ▁to ▁higher ▁ram ification ▁sub groups , ▁see ▁e . g . ▁Ch . ▁IV ▁of . ▁ ▁Using ▁the ▁local ▁reci pro city ▁map , ▁one ▁defines ▁the ▁Hil bert ▁symbol ▁and ▁its ▁general izations . ▁F inding ▁explicit ▁formulas ▁for ▁it ▁is ▁one ▁of ▁sub dire ctions ▁of ▁the ▁theory ▁of ▁local ▁fields , ▁it ▁has ▁a ▁long ▁and ▁rich ▁history , ▁see ▁e . g . ▁Serge i ▁V ost ok ov ' s ▁review . ▁ ▁There ▁are ▁coh om ological ▁approaches ▁and ▁non - co hom ological ▁approaches ▁to ▁local ▁class ▁field ▁theory . ▁C oh om ological ▁approaches ▁tend ▁to ▁be ▁non - exp licit , ▁since ▁they ▁use ▁the