Let $K$ be a charateristic $p$ global field. For each place $P$ we let $K_P$ to be the completion at $P$ (which is a local field) and let $t_P$ be their prime.
Let $t$ be a separating element of $K$ (which means $K/F(t)$ is a separable extension where $F$ is the constant field over $K$).
Then Artin-Tate claims that $\frac{dt}{dt_P}$ is almost always a local unit. ie for almost all $P$ $v_P(\frac{dt}{dt_P})=0$.
It is clear that $v_P(t)=0$ for almost all $t$ so I just need to verify that for each expansion of $t$ at $k_P$, say
$$t=\sum_{i\geq0}a_i t_P^{i},$$ $a_1$ is almost always non zero.
But I have absolutely no clue how to prove it.
The book refers to chapter 17-4 of Algebraic numbers and algebraic functions but I couldn't get anything out of it.
Any help would be great!