Timeline for Order of roots for a polynomial $P\in\mathbb{F}_p[T]$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 22, 2022 at 21:47 | vote | accept | Gabriel Soranzo | ||
| Apr 22, 2022 at 21:14 | comment | added | Will Sawin | @GabrielSoranzo Ah, I see what has happened. When you said "all roots have a certain order $k$" I assumed you meant the same order $k$ for all roots. My answer was under that assumption. For the question of simply what numbers can be the order of the roots of a polynomial of degree $d$, you indeed want to consider the divisors of $p^k-1$ for $k \leq \deg P$. | |
| Apr 22, 2022 at 19:58 | comment | added | Gabriel Soranzo | Following your answer I could consider the divisor of $p^k-1$ for $k\leq\deg(P)$? | |
| Apr 22, 2022 at 19:56 | comment | added | Gabriel Soranzo | Yes: I do not supposed that $P$ is irreducible! I need such a bound for a factorization algorithm | |
| Apr 22, 2022 at 19:54 | comment | added | Max Alekseyev | $P=X^5-1$ is not irreducible. | |
| Apr 22, 2022 at 19:35 | comment | added | Gabriel Soranzo | With $P=X^5-1$ in $\mathbb{F}_3$ the orders are the divisors of $5$ but $p^d-1=2\times 11^2$ so $5$ doesnt divide $p^d-1$?? I missed something? | |
| Apr 22, 2022 at 19:31 | comment | added | Will Sawin | @GabrielSoranzo That's a list of possible orders. Not all possibilities necessarily occur for all polynomials of a given degree. | |
| Apr 22, 2022 at 19:29 | comment | added | Gabriel Soranzo | If I take $\Phi_4=X^2+1$ (for example over $\mathbb{F}_3$) then $p^d-1=3^2-1=8$. But all the roots of $\Phi_4$ have order 4, and we have also $8$, $2$ and $1$ for divisors of $8$? What's the matter? | |
| Apr 22, 2022 at 15:37 | history | answered | Will Sawin | CC BY-SA 4.0 |