▁that ▁provides ▁a ▁measure ▁of ▁size ▁or ▁multip licity ▁of ▁elements ▁of ▁the ▁field . ▁It ▁general izes ▁to ▁commut ative ▁algebra ▁the ▁notion ▁of ▁size ▁inher ent ▁in ▁consideration ▁of ▁the ▁degree ▁of ▁a ▁pole ▁or ▁multip licity ▁of ▁a ▁zero ▁in ▁complex ▁analysis , ▁the ▁degree ▁of ▁divis ibility ▁of ▁a ▁number ▁by ▁a ▁prime ▁number ▁in ▁number ▁theory , ▁and ▁the ▁geomet rical ▁concept ▁of ▁contact ▁between ▁two ▁algebraic ▁or ▁analyt ic ▁vari eties ▁in ▁algebraic ▁geometry . ▁A ▁field ▁with ▁a ▁valu ation ▁on ▁it ▁is ▁called ▁a ▁val ued ▁field . ▁ ▁Definition ▁ ▁One ▁starts ▁with ▁the ▁following ▁objects : ▁a ▁field ▁ ▁and ▁its ▁multip lic ative ▁group ▁K × , ▁an ▁ab elian ▁totally ▁ordered ▁group ▁. ▁The ▁ordering ▁and ▁group ▁law ▁on ▁ ▁are ▁extended ▁to ▁the ▁set ▁} ▁by ▁the ▁rules ▁ ▁for ▁all ▁▁ ∈ ▁, ▁ ▁for ▁all ▁▁ ∈ ▁ ▁. ▁ ▁Then ▁a ▁valu ation ▁of ▁ ▁is ▁any ▁map ▁ ▁which ▁satisfies ▁the ▁following ▁properties ▁for ▁all ▁a , ▁b ▁in ▁K : ▁▁ ▁if ▁and ▁only ▁if ▁, ▁, ▁, ▁with ▁equality ▁if ▁v ( a ) ▁ ≠ ▁v ( b ). ▁ ▁A ▁valu ation ▁v ▁is ▁trivial ▁if ▁v ( a )   =   0 ▁for ▁all ▁a ▁in ▁K × , ▁otherwise ▁it ▁is ▁non - t rivial . ▁ ▁The ▁second ▁property ▁as ser ts ▁that ▁any ▁valu ation ▁is ▁a ▁group ▁hom omorphism . ▁The ▁third ▁property ▁is ▁a ▁version