I'm trying to understand why if the $gcd(p-1, 15) = d \neq 15$, then there are zero roots (since if it's $=15$, there are exactly 8). I was thinking that since a solution to $x^d - 1$ is relevant if $gcd(p-1, 15) = d \neq 15$, wouldn't the roots of $\Phi_{15}[x]$ technically still appear in $\mathbb F_p$ for $p = 1, 3, 5$? Why is there no in between and simple 8 solutions or none?
2026-03-30 12:01:48.1774872108
Number of roots in cyclotomic polynomial $\Phi_{15}[x]$ in $\mathbb F_p$
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