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Let $p$ be a prime number. $k=\bar{\mathbb{F}}_p$. Let $f=x+t\in \mathbb{F}_q$, which is irreducible in $x$ and separable in $t$. Here, $x$ is substituted as $x=\sum_{i=0}^{n}a_it^i\in \mathbb{F}_p[t][a_0,...,a_n]$, so that, $f(a_0,...a_n,t)=\sum_{i=0}^{n}a_it^i$ be separable over $k(a_0,...,a_n)$, and $a_i$ are algebraically independent variables over $\mathbb{F}_p$.

Does this mean, that $Disc_{t}f(\alpha)$ is not zero for some $\alpha$ in k. Any hint or method is of great help.

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  • $\begingroup$ What exactly is $\;G\;$ ?? The sum of a polynomial and a polynomial ring? What is that? $\endgroup$
    – DonAntonio
    May 8, 2018 at 9:26
  • $\begingroup$ We are in function field case. $G$ is in $k[t][x].$, $x$ is a polynomial in t. and by assumption, $G$ is irreducible in $x$ and separable in t $\endgroup$
    – Math123
    May 8, 2018 at 9:36
  • $\begingroup$ What is the meaning of $\;g(t)+F[t,x]\;$ ? Thsi is what I am asking $\endgroup$
    – DonAntonio
    May 8, 2018 at 9:37
  • $\begingroup$ Ok, it can be some thing of this type: $g(t)+x$, after substitution for x, it looks like: $g(t)+\sum_{i=0}^{i=n}a_it^i$ $\endgroup$
    – Math123
    May 8, 2018 at 9:39

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