▁polynomial ▁ ▁with ▁, ▁define ▁va ( f ) ▁= ▁k , ▁the ▁order ▁of ▁van ishing ▁at ▁x ▁= ▁a ; ▁and ▁va ( f ▁/ g ) ▁= ▁va ( f ) ▁− ▁va ( g ). ▁Then ▁the ▁valu ation ▁ring ▁R ▁consists ▁of ▁rational ▁functions ▁with ▁no ▁pole ▁at ▁x ▁= ▁a , ▁and ▁the ▁completion ▁is ▁the ▁formal ▁Laurent ▁series ▁ring ▁F (( x − a )). ▁This ▁can ▁be ▁generalized ▁to ▁the ▁field ▁of ▁Pu ise ux ▁series ▁K {{ t }} ▁( f raction al ▁powers ), ▁the ▁Le vi - C iv ita ▁field ▁( its ▁C auch y ▁completion ), ▁and ▁the ▁field ▁of ▁H ahn ▁series , ▁with ▁valu ation ▁in ▁all ▁cases ▁returning ▁the ▁smallest ▁exponent ▁of ▁t ▁appearing ▁in ▁the ▁series . ▁ ▁- adic ▁valu ation ▁ ▁General izing ▁the ▁previous ▁examples , ▁let ▁ ▁be ▁a ▁principal ▁ideal ▁domain , ▁ ▁be ▁its ▁field ▁of ▁fra ctions , ▁and ▁ ▁be ▁an ▁ir reducible ▁element ▁of ▁. ▁Since ▁every ▁principal ▁ideal ▁domain ▁is ▁a ▁unique ▁factor ization ▁domain , ▁every ▁non - zero ▁element ▁a ▁of ▁ ▁can ▁be ▁written ▁( ess entially ) ▁uniqu ely ▁as ▁ ▁where ▁the ▁es ▁are ▁non - negative ▁integers ▁and ▁the ▁pi ▁are ▁ir reducible ▁elements ▁of ▁ ▁that ▁are ▁not ▁associ ates ▁of ▁. ▁In ▁particular , ▁the ▁integer ▁e a ▁is ▁uniqu ely ▁determined ▁by ▁a . ▁ ▁The ▁ π - adic ▁valu ation ▁of ▁K ▁is ▁then ▁given