▁construct ▁its ▁completion , ▁as ▁in ▁the ▁examples ▁below , ▁and ▁different ▁valu ations ▁can ▁define ▁different ▁completion ▁fields . ▁ ▁In ▁general , ▁a ▁valu ation ▁indu ces ▁a ▁uniform ▁structure ▁on ▁, ▁and ▁ ▁is ▁called ▁a ▁complete ▁val ued ▁field ▁if ▁it ▁is ▁complete ▁as ▁a ▁uniform ▁space . ▁There ▁is ▁a ▁related ▁property ▁known ▁as ▁spher ical ▁comple ten ess : ▁it ▁is ▁equivalent ▁to ▁comple ten ess ▁if ▁ ▁but ▁stronger ▁in ▁general . ▁ ▁Ex amples ▁ ▁p - adic ▁valu ation ▁The ▁most ▁basic ▁example ▁is ▁the ▁- adic ▁valu ation ▁v p ▁associated ▁to ▁a ▁prime ▁integer ▁p , ▁on ▁the ▁rational ▁numbers ▁ ▁with ▁valu ation ▁ring ▁ ▁The ▁valu ation ▁group ▁is ▁the ▁add itive ▁integers ▁ ▁For ▁an ▁integer ▁ ▁the ▁valu ation ▁v p ( a ) ▁measures ▁the ▁divis ibility ▁of ▁a ▁by ▁powers ▁of ▁p : ▁ ▁and ▁for ▁a ▁fraction , ▁v p ( a / b ) ▁= ▁v p ( a ) ▁− ▁v p ( b ). ▁ ▁Writing ▁this ▁multip lic atively ▁yields ▁the ▁- adic ▁absolute ▁value , ▁which ▁convention ally ▁has ▁as ▁base ▁, ▁so ▁. ▁ ▁The ▁completion ▁of ▁ ▁with ▁respect ▁to ▁v p ▁is ▁the ▁field ▁ ▁of ▁p - adic ▁numbers . ▁ ▁Order ▁of ▁van ishing ▁Let ▁K ▁= ▁F ( x ), ▁the ▁rational ▁functions ▁on ▁the ▁aff ine ▁line ▁X ▁= ▁F 1 , ▁and ▁take ▁a ▁point ▁a ▁ ∈ ▁X . ▁For ▁a