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 $$ \max_{\deg(a)=1}r(x^i-a)=r(x^i-x) $$
2026-04-11 22:01:55.1775944915
Maximal Distinct Roots in $F_q$
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Hints: Show that for every $a,b \in F_q$,
$r(x^i - ax) \le r(x^i - x)$.
$r(x^i - ax - b) \le r(x^i - ax)$.
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.