I am studying finite field theory and Galois theory from NPTEL lecture series on Galois Theory. Here I found an example and I ahve question related to that. The example is about an irreducible inseparable polynomial. Here it is :
Let us consider the field $\Bbb F_p (t)$ of rational functions over $\Bbb F_p$ with variable $t.$ Let us consider the polynomial $f(X)=X^p - t \in \Bbb F_p (t) [X].$ Then it is irreducible over $\Bbb F_p (t)$ but it is inseparable since $f'(X) = pX^{p-1} = 0$ in $\Bbb F_p(t)[X]$ as $\text{Char}\ (\Bbb F_p(t)) = p.$
Inseparability can be shown in other ways too. Let $x$ be a zero of $f(X)$ in some extension field. Then $x$ is algebraic over $\Bbb F_p(t).$ But then $\Bbb F_p(t)[x]$ is a field i.e. $\Bbb F_p(t)[x]=\Bbb F_p(t)(x).$
Now $\text{Char}\ (\Bbb F_p(t)(x))=p.$ So we have $$f(X)=X^p-t=X^p-x^p = (X-x)^p\ \text{in}\ \Bbb F_p(t)(x)[X].$$
So $x$ is a zero of $f(X)$ of multiplicity $p$ i.e. $f(X)$ is not separable.
My question is $:$ Is $f(X)$ not irreducible in $\Bbb F_p[t][X] = \Bbb F_p[t,X]$?
I have an idea regarding this. What we know is that a polynomial in $\Bbb Z[X]$ is irreducible in $\Bbb Z[X]$ iff it is irreducible in $\Bbb Q[X]$ and it is a primitive polynomial, where $\Bbb Q$ is the quotient field of $\Bbb Z.$ So what I think is that in our case $t$ is prime in $\Bbb F_p[t].$ So by Eisenstein's criterion it actually follows that $X^p-t$ is irreducible over the quotient field $\Bbb F_p[t]$ which is $\Bbb F_p(t).$ Also since $X^p-t$ is a primitive polynomial so it is irreducible in $\Bbb F_p[t][X] = \Bbb F_p[t,X].$ Am I correct in my argument?
Any valuable suggestion regarding this will be highly appreciated. Thank you very much.