▁by ▁ ▁If ▁ π ' ▁is ▁another ▁ir reducible ▁element ▁of ▁ ▁such ▁that ▁( π ')   =   ( π ) ▁( that ▁is , ▁they ▁generate ▁the ▁same ▁ideal ▁in ▁R ), ▁then ▁the ▁ π - adic ▁valu ation ▁and ▁the ▁ π '- adic ▁valu ation ▁are ▁equal . ▁Thus , ▁the ▁ π - adic ▁valu ation ▁can ▁be ▁called ▁the ▁P - adic ▁valu ation , ▁where ▁P   =   ( π ). ▁▁ ▁P - adic ▁valu ation ▁on ▁a ▁D ed ek ind ▁domain ▁The ▁previous ▁example ▁can ▁be ▁generalized ▁to ▁D ed ek ind ▁domains . ▁Let ▁ ▁be ▁a ▁D ed ek ind ▁domain , ▁ ▁its ▁field ▁of ▁fra ctions , ▁and ▁let ▁P ▁be ▁a ▁non - zero ▁prime ▁ideal ▁of ▁. ▁Then , ▁the ▁local ization ▁of ▁ ▁at ▁P , ▁denoted ▁R P , ▁is ▁a ▁principal ▁ideal ▁domain ▁whose ▁field ▁of ▁fra ctions ▁is ▁. ▁The ▁construction ▁of ▁the ▁previous ▁section ▁applied ▁to ▁the ▁prime ▁ideal ▁PR P ▁of ▁R P ▁yields ▁the ▁- adic ▁valu ation ▁of ▁. ▁▁ ▁Ge ometric ▁notion ▁of ▁contact ▁ ▁Val u ations ▁can ▁be ▁defined ▁for ▁a ▁field ▁of ▁functions ▁on ▁a ▁space ▁of ▁dimension ▁greater ▁than ▁one . ▁Rec all ▁that ▁the ▁order - of - van ishing ▁valu ation ▁va ( f ) ▁on ▁ ▁measures ▁the ▁multip licity ▁of ▁the ▁point ▁x ▁= ▁a ▁in ▁the ▁zero ▁set ▁of ▁f ; ▁one ▁may ▁consider ▁this