Fundamental Theorem of Galois Theory - why does my book have different assumptions?

Finite fields and fields of characteristic zero are examples of perfect fields, which have the property that every finite extension is separable. For fields of characteristic zero this is fairly clear, while for finite fields the important point is that the Frobenius endomorphism $x\mapsto x^p$ is surjective.

So the statement in your book is less general, and was likely chosen to avoid dealing with separability.

In a finite field of order $q$, the product of all monic irreducible polynomials of degree dividing $n$ is $x^{q^n}-x$. Therefore, if you prove that $x^{q^n} - x$ is separable, then so is any minimal polynomial.

You can show that a polynomial $f(x)$ has a repeated root at $x = a$ if and only if both $f(a)=0$ and $f'(a) = 0$ (here we define the derivative of a polynomial formally to be what we know it should be from calculus).

Since the derivative of $x^{q^n} - x$ is $-1$, $x^{q^n} -x $ has no repeated roots. I.e. it is separable.