I am reading the book "Algebraic Geometry in Coding Theory and Cryptography" by NIEDERREITER & XING. I have a doubt in last paragraph of page 23 & 1st paragraph of page 24. Please help me.
Let $P$ be a place of $F /k$ and let $t \in F$ be a local parameter of $F$ at place $P$. Given an element $z \in F$ , we choose an integer $r$ such that valuation $v_P (z) \geq r$. Then $zt^{−r} \in O_P$ , and so we can put $a_r=(zt^{-r})(P)$. Then how $z-a_rt^r \in F$ to define valuation $v_P$ over it?