Field extension generated by $\alpha$ and separability

Question

In my notes, I have written that the field extension of $k$ generated by an element $\alpha \in K$, where $K$ a larger field, is defined to be $$k(\alpha) = \bigcap_{ \alpha \in E} E$$ where $E \subset K$ are subfields of $K$, and $k\subset E$.

A $\textbf{separable}$ field extension is a field extension $K$ over a smaller field $k$ such that every element of $K$ is a root of a separable polynomial over $k$, let $k$ be a finite field.

Let us consider $f(x) \in k[x]$, irreducible, then $f(x)$ is separable. Now let $\alpha \in K$ be a root of $f(x)=0$ and let us generate the finite field extension $k(\alpha)$.

In my notes I also have that all finite extensions of finite fields are separable. So in this case, $k(\alpha)$ is finite extension of a finite field and thus should be separable.

I know that, of course, for $\alpha$, there exists a separable polynomial over $k$, namely the irreducible $f(x)$. However, how do we know that ALL of the elements of $k(\alpha)$ are the roots of separable polynomials in $k[x]$ ?

Overall, how are the elements of $k(\alpha)$ related to $\alpha$?

Answer 1

All the elements of $k(\alpha)$ are separable, because all irreducible polynomials over a finite field are separable. Indeed if $f(x)$ is not separable, $ f(x)$ and $f'(x)$ have a common root in some extension of $k$, hence $\deg\gcd(f(x),f'(x))\ge 1$, which is impossible if $f$ is irreducible, unless $f(x)=g(x^p)$, for some polynomial $g$ (in which case $f'(x)=p x^{p-1}g'(x)=0$).

But the Frobenius morphism $x\mapsto x^p$ is injective, hence surjective, since we're in a finite field, so that $f(x)=g(x^p)=\bigl(h(x)\bigr)^p$ for some $h(x)\in k[x]$, and $f(x)$ would not be irreducible.