math -- elements of Z_p[x] which permute Z_p
- From: quasi <quasi@xxxxxxxx>
- Date: Fri, 11 Apr 2008 19:24:57 -0500
The following is easily proved:
If p is prime, then for every function F: Z_p --> Z_p, there exists a
unique polynomial f in Z_p[x], with deg(f) < p, such that, regarding f
as a function from Z_p to Z_p, we have F = f.
In particular, every element of the permutation group S_p is uniquely
represented by a polynomial in Z_p[x] with degree < p.
The simplest examples are the linear polynomials, which form a
subgroup (under composition) of order p^2 - p.
With that as background, here's a conjecture ...
Conjecture:
If p is prime, p > 2, there does not exist f in Z_p[x], with deg(f) =
p - 1, which permutes Z_p.
Remarks:
While I've verified the conjecture for p = 3, 5, 7, 11, I don't have a
lot of confidence that it holds for all primes p > 2. I would estimate
my degree of belief as only about 60-40, based on what I know at this
point.
quasi
.
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