Maximal Distinct Roots in $F_q$

Question

Let $a\in F_q[x]$, and let $r(\cdot)$ denote the number of distinct roots over $F_q$. For any $i|q$, prove that $$ \max_{\deg(a)=1}r(x^i-a)=r(x^i-x) $$

Answer 1

Hints: Show that for every $a,b \in F_q$,

1. $r(x^i - ax) \le r(x^i - x)$.

2. $r(x^i - ax - b) \le r(x^i - ax)$.

For 1), factor out an $x$, then use the fact that the roots of $x^{i-1}-1$ form a multiplicative subgroup. For 2), use the fact that the roots of $x^i - ax$ form an additive subgroup.