▁length ▁of ▁ ▁This ▁length ▁is ▁finite , ▁and ▁it ▁is ▁add itive ▁with ▁respect ▁to ▁multiplication , ▁that ▁is , ▁. ▁If ▁k ( X ) ▁is ▁the ▁field ▁of ▁rational ▁functions ▁on ▁X , ▁then ▁any ▁non - zero ▁ ▁may ▁be ▁written ▁as ▁a ▁quot ient ▁, ▁where ▁g ▁and ▁h ▁are ▁in ▁ ▁and ▁the ▁order ▁of ▁van ishing ▁of ▁f ▁is ▁defined ▁to ▁be ▁. ▁With ▁this ▁definition , ▁the ▁order ▁of ▁van ishing ▁is ▁a ▁function ▁. ▁If ▁X ▁is ▁normal , ▁then ▁the ▁local ▁ring ▁ ▁is ▁a ▁discrete ▁valu ation ▁ring , ▁and ▁the ▁function ▁ ▁is ▁the ▁corresponding ▁valu ation . ▁For ▁a ▁non - zero ▁rational ▁function ▁f ▁on ▁X , ▁the ▁principal ▁We il ▁divis or ▁associated ▁to ▁f ▁is ▁defined ▁to ▁be ▁the ▁We il ▁divis or ▁ ▁It ▁can ▁be ▁shown ▁that ▁this ▁sum ▁is ▁locally ▁finite ▁and ▁hence ▁that ▁it ▁indeed ▁defines ▁a ▁We il ▁divis or . ▁The ▁principal ▁We il ▁divis or ▁associated ▁to ▁f ▁is ▁also ▁not ated ▁. ▁If ▁f ▁is ▁a ▁regular ▁function , ▁then ▁its ▁principal ▁We il ▁divis or ▁is ▁effective , ▁but ▁in ▁general ▁this ▁is ▁not ▁true . ▁The ▁add itivity ▁of ▁the ▁order ▁of ▁van ishing ▁function ▁implies ▁that ▁ ▁Con sequently ▁ ▁is ▁a ▁hom omorphism , ▁and ▁in ▁particular ▁its ▁image ▁is ▁a ▁subgroup ▁of ▁the ▁group ▁of ▁all ▁We il ▁divis ors . ▁ ▁Let ▁X ▁be ▁a ▁normal ▁integral ▁No ether ian ▁scheme . ▁Every