GAP Manual: 18.3. Operations for Finite Field Elements

z1 + z2
z1 - z2
z1 * z2
z1 / z2

The operators +, -, * and / evaluate to the sum, difference, product, and quotient of the two finite field elements z1 and z2, which must lie in fields of the same characteristic. For the quotient / z2 must of course be nonzero. The result must of course lie in a finite field of size less than or equal to 2^(16), otherwise an error is signalled.

Either operand may also be an integer i. If i is zero it is taken as the zero in the finite field, i.e., F.zero, where F is a field record for the finite field in which the other operand lies. If i is positive, it is taken as i-fold sum F.one+F.one+..+F.one. If i is negative it is taken as the additive inverse of -i.

    gap> Z(8) + Z(8)^4;
    Z(2^3)^2
    gap> Z(8) - 1;
    Z(2^3)^3
    gap> Z(8) * Z(8)^6;
    Z(2)^0
    gap> Z(8) / Z(8)^6;
    Z(2^3)^2
    gap> -Z(9);
    Z(3^2)^5 

z ^ i

The powering operator ^ returns the i-th power of the element in a finite field z. i must be an integer. If the exponent i is zero, z^i is defined as the one in the finite field, even if z is zero; if i is positive, z^i is defined as the i-fold product z*z*..*z; finally, if i is negative, z^i is defined as (1/z)^-i. In this case z must of course be nonzero.

    gap> Z(4)^2;
    Z(2^2)^2
    gap> Z(4)^3;
    Z(2)^0    # is in fact 1
    gap> (0*Z(4))^0;
    Z(2)^0