First, we define an univariate polynomial:
poly: x^6 +19*x^5 + x^4 - 14*x^3 - x^2 - 3*x + 1;
6 5 4 3 2
x + 19 x + x - 14 x - x - 3 x + 1
To convert this polynomial into one with coefficients that are rests modulo 11, we use the function mod:
mod(poly, 11);
And we get this result:
6 5 4 3 2
x - 3 x + x - 3 x - x - 3 x + 1
if set to a positive prime p, then all arithmetic in the rational function routines will be done modulo p. That is all integers will be reduced to less than p/2 in absolute value (if p=2 then all integers are reduced to 1 or 0). This is the so called "balanced" modulus system, e.g. N MOD 5 = -2, -1, 0, 1, or 2.
To factor the polynomial poly modulo 11, we have to add the option modulus:11 to the command factor:
factor(poly), modulus:11;
and obtain
2 3 2 (x + 1) (x + 5 x + 3) (x + 2 x + 3 x + 4)
Expansion of that product is performed over the integers:
expand(%)
6 5 4 3 2 x + 8 x + 23 x + 41 x + 54 x + 41 x + 12
Reduction modulo 11 gives the polynomial that we obtained by reduction of poly:
mod(%, 11)
6 5 4 3 2 x - 3 x + x - 3 x - x - 3 x + 1