Are $\mathbb{F}_{p^{d-1}} \subset \mathbb{F}_{p^d}$ Galois Extensions?

Question

I know that $\mathbb{F}_{p}\subset \mathbb{F}_{p^d}$ is Galois extensions and the degree of extension $[\mathbb{F}_{p^d} : \mathbb{F}_{p}]=d$.

but my question is :-

1. $\mathbb{F}_{p^{d-1}} \subset \mathbb{F}_{p^d}$ is Galois Extensions
2. $\mathbb{F}_{p^k} \subset \mathbb{F}_{p^d}$ is Galois Extensions

in both case $p$ is prime and $d,k$ are integers ( also $ k>d$).

Answer 1

$\mathbf F_{p^k}\subset\mathbf F_{p^d}$ if and only if $k\mid d$.

And, yes, in that case, the extension is Galois. This is because the Galois group $\;\operatorname{Gal}_{\mathbf F_{p^d}/\mathbf F_p}$ is abelian, isomorphic to $\mathbf Z/d\mathbf Z$, hence all its subgroups are normal.