= 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 $(xy)^q=x^qy^q$. 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 $d$.
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