 Solution

Let $F$ be the splitting field over ${\mathbb{F}}_p$ of $X^q-X$. Its derivative is $-1$, so it has $q$ distinct roots. The roots are closed under addition, subtraction, multiplication, and inversion, using $(x+y{)}^q=x^q+y^q$ and $(xy{)}^q=x^q{y}^q$. They therefore form a field with $q$ elements.

Every field with $q$ elements has multiplicative group of order $q-1$, so every element satisfies $x^q=x$; it is therefore a splitting field of the same polynomial and is unique up to isomorphism. If an irreducible factor over ${\mathbb{F}}_p$ has a root $\alpha \in {\mathbb{F}}_q$, its degree is $[{\mathbb{F}}_p(\alpha ):{\mathbb{F}}_p]$, which divides $d$ by the tower law. In particular it is at most $d$.