Splitting fields of polynomials over finite fields

Solution 1:

i) Your edit is correct. The statement is telling you that any finite extension of a finite field is normal. Since the splitting field of $f$ is necessarily finite (with degree at most $d!$), then it must also be normal.

ii) $K(\alpha)$ is a finite extension of $K$, and so (by the previous sentence) is a normal extension. It contains a root of $f$, so must therefore contain all the roots of $f$ by normality, and therefore contains the splitting field of $f$. To see that $K(\alpha)$ is precisely the splitting field of $f$, observe that a splitting field of $f$ must contain $K$ and $\alpha$ and so contains $K(\alpha)$.