Let $K$ be a function field of positive characteristic $p$.
Let $b$, $u$, $u'$ be in $K\setminus K^p$. 
I would like to show that there are only finitely many natural numbers $r$ prime to $p$ such that for some natural number $n\_r$ we have $u'=(b^r)(u^{p^{n\_r}})$.
But I don't even know whether this assertion holds.



A question in a function field of positive characteristic $p$
That is not true. Let $K$ be $\mathbb{F}\_p(u)$, a purely transcendental extension of degree $1$. Let $u'$ equal $u$. Let $b$ equal $u^{1-p}$. None of these is in $K^p$. For every integer $n\geq 0$, for the positive natural number $r=(p^n-1)/(p-1)$, then $b^ru^{p^n}$ equals $u'$. Of course $r$ is congruent to $1$ modulo $p$, so $r$ is prime to $p$.


