Arithmetic over finite fields (Galois fields)

A special representation for univariate polynomials over finite fields defined by a single algebraic extension over Z mod p has been implemented. The new facility includes support for computing polynomial gcd's, factoring, and finding roots. It is accessed via the mod function. Example: factor the polynomial over then .

    > f := x^6+x^5+x^4+x^3+1:
    > Factor(f) mod 2;

                             2            4
                           (x  + x + 1) (x  + x + 1)

    > alias(alpha=RootOf(x^2+x+1)):
    > Factor(f,alpha) mod 2;

                    2                                    2
      (x + alpha) (x  + x + alpha + 1) (x + alpha + 1) (x  + x + alpha)

An algebraic extension can also be given to the Roots and Irreduc functions, e.g.

    > Roots(f,alpha) mod 2;

                          [[alpha, 1], [alpha + 1, 1]]