Regarding valuation on function field

Regarding valuation on function field

Notations: Let $k$ be a field of characteristic $\neq 2$, $k(X)$ be a
function field. Suppose $p(X)\in k[X]$ is an irreducible polynomial and
$\alpha$ be a root of $p(X)$. Let $F=k(\alpha)=k[X]/p(X).k[X]$. We can
define $p(X)$-adic valuation on $k(X)$ as : for $a(X)\in k(X)$, it can be
written as $a(X)=p(X)^n.\frac{h(X)}{r(X)}$, for some $n\in\mathbb Z$
,$(h(X),r(X))=1$ and $p(X)$ does not divide $r(X)$. Then $p(X)$-adic
valution of $a(X)$ is $n$.
Question: If $g(X)\in k(X)$ is such that $p(X)$-adic valuation of $g(X)$
is odd. Then we need to show that $(X-\alpha)$-addic valuation (this is
over $F(X)$) of $g(X)$ is also odd.
Thanks!