▁to ▁v , ▁the ▁class ▁of ▁v ▁under ▁the ▁equivalence ▁defined ▁below . ▁ ▁Basic ▁properties ▁ ▁Equ ival ence ▁of ▁valu ations ▁ ▁Two ▁valu ations ▁v 1 ▁and ▁v 2 ▁of ▁ ▁with ▁valu ation ▁group ▁ Γ 1 ▁and ▁ Γ 2 , ▁respectively , ▁are ▁said ▁to ▁be ▁equivalent ▁if ▁there ▁is ▁an ▁order - pres erving ▁group ▁isomorphism ▁ ▁such ▁that ▁v 2 ( a )   =   φ ( v 1 ( a )) ▁for ▁all ▁a ▁in ▁K × . ▁This ▁is ▁an ▁equivalence ▁relation . ▁ ▁Two ▁valu ations ▁of ▁K ▁are ▁equivalent ▁if ▁and ▁only ▁if ▁they ▁have ▁the ▁same ▁valu ation ▁ring . ▁ ▁An ▁equivalence ▁class ▁of ▁valu ations ▁of ▁a ▁field ▁is ▁called ▁a ▁place . ▁Ost row ski ' s ▁theorem ▁gives ▁a ▁complete ▁classification ▁of ▁places ▁of ▁the ▁field ▁of ▁rational ▁numbers ▁ ▁these ▁are ▁precisely ▁the ▁equivalence ▁classes ▁of ▁valu ations ▁for ▁the ▁p - adic ▁complet ions ▁of ▁ ▁Ext ension ▁of ▁valu ations ▁Let ▁v ▁be ▁a ▁valu ation ▁of ▁ ▁and ▁let ▁L ▁be ▁a ▁field ▁extension ▁of ▁. ▁An ▁extension ▁of ▁v ▁( to ▁L ) ▁is ▁a ▁valu ation ▁w ▁of ▁L ▁such ▁that ▁the ▁restriction ▁of ▁w ▁to ▁ ▁is ▁v . ▁The ▁set ▁of ▁all ▁such ▁extensions ▁is ▁studied ▁in ▁the ▁ram ification ▁theory ▁of ▁valu ations . ▁ ▁Let ▁L / K ▁be ▁a ▁finite ▁extension ▁and ▁let ▁w ▁be ▁an ▁extension ▁of ▁v ▁to ▁L . ▁The ▁index