[seqfan] Re: x^k-1 divisible by an irreducible polynomial in a finite field. A371164 and A106309

[Edwin Clark]

On Tue, Mar 26, 2024 at 7:25 PM <israel at math.ubc.ca> wrote:

>
> Suppose the monic polynomial q(z) of degree d >1 is irreducible over a
> finite field F (the integers modulo a prime p, if that makes a
> difference).
> I want to find the least positive integer k such that z^k - 1 is divisible
> by q(z) over F. If I take the extension field K = F[r] where r is a root
> of
> q(z), is the answer the order of r in the multiplicative group K^x$ of K?
>

This is Theorem 3.3 in Finite Fields (Encyclopedia of Mathematics and
its Applications) by Lidl and Niederreiter (page 84)
provided q(0) is not 0.