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
GAP 3.4.4