ConwayPolynomial( p, n )
returns the Conway polynomial of the finite field GF(p^n) as polynomial over the Rationals.
The Conway polynomial Phi_{n,p} of GF(p^n) is defined by the following properties.
First define an ordering of polynomials of degree n over GF(p) as follows.
f = sum_{i=0}^n (-1)^i f_i x^i is smaller than g = sum_{i=0}^n (-1)^i g_i x^i if and only if there is an index m leq n such that f_i = g_i for all i > m, and tilde{f_m} < tilde{g_m}, where tilde{c} denotes the integer value in { 0, 1, ldots, p-1 } that is mapped to c in GF(p) under the canonical epimorphism that maps the integers onto GF(p).
Phi_{n,p} is primitive over GF(p), that is, it is irreducible, monic, and is the minimal polynomial of a primitive element of GF(p^n) over GF(p).
For all divisors d of n the compatibility condition Phi_{d,p}( x^{frac{p^n-1}{p^m-1}} ) equiv 0 pmod{Phi_{n,p}(x)} holds.
With respect to the ordering defined above, Phi_{n,p} shall be minimal.
gap> ConwayPolynomial( 7, 3 );
X(Rationals)^3 + 6*X(Rationals)^2 + 4
gap> ConwayPolynomial( 41, 3 );
X(Rationals)^3 + X(Rationals) + 35
The global list CONWAYPOLYNOMIALS contains Conway polynomials for small
values of p and n.
Note that the computation of Conway polynomials may be very expensive,
especially if n is not a prime.
GAP 3.4.4