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

[israel at math.ubc.ca]

This is relevant to A371164 and A106309. I think this is correct, but I'd 
like confirmation from someone better acquainted with finite fields than I 
am.

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?

For example, take F to be the integers mod 13, and 
q(z) = x^4 + 5 x^2 + x + 10.  
Maple (using the GF package) tells me the order of r in this case is
14280.  Indeed z^{14280}-1 is divisible by q(z) over F in this case, 
and no divisor of 14280 will work.

Cheers,
Robert