Preparing for my Galois theory exam in may and i am faced with the following question.
Give an example of a finite inseparable extension with a sketched proof of its inseparability
I have the following proposition
Proposition Suppose $f ∈ F[X]$ is irreducible and inseparable over $F$. Then $char F = p,$ for some prime number $p$, and there is a polynomial $g ∈ F[X]$ such that $f(X) = g(X^{p}).$
Which makes me think of the frobenius map
$\sigma : F \mapsto F$
$x \mapsto x^{p} $
where $F$ is a field of characteristic $p$ prime (note this map is a homomorphism)
However i dont really know how to use any of this, or even if i am on the right lines for finding a finite inseparable extension.