Most statements of the Fundamental Theorem of Galois Theory I've seen online state it for finite, Galois extensions. A Galois extension is defined as one that is both separable and normal. So most authors would begin the theorem with something like this:
Let $F$ be a field and $E$ a finite, normal, separable extension of $F$.
My text states:
Let $F$ be either a finite field or a field of characteristic $0$. Suppose $E$ is a finite, normal extension of $F$.
Are these equivalent? I'm having a hard time seeing this.
If $F$ has characteristic $0$: consider any $a \in E.$ Since $E$ is normal it is also algebraic, so that there is some (irreducible) minimal polynomial $f_a(x)$ over $F$ such that $f_a(a) = 0.$ Since $F$ has characteristic $0$, we know that an irreducible polynomial is separable. Thus, for every $a \in E$ we know there is a separable $f_a(x)$ such that $f_a(a) = 0$, and so $E$ must be a separable extension.
If $F$ is a finite field: note the characteristic of $F$ is some prime $p$. The same general argument applies but for finite fields, it is not true that any irreducible polynomial $f(x)$ is separable. Instead we have the additional condition that $f(x)$ is not of the form $g(x^p).$
If we can show that the minimal polynomial $f(x)$ is not of the form $g(x^p)$, then this would prove that $E$ is a separable extension. I'm also having trouble showing the other direction.