\\ The following computes the square of the Gauss sum over Q, attached to a prime number p.

p = 13;
Z = Mod(x, polcyclo(p));
print(Z);
Mod(x, x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
S = sum(j = 0, p - 1, kronecker(j, p)*Z^j);
print((S^2));
Mod(13, x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
print(lift(S^2));
13


\\ The following gives the factorisation of the reduction at a prime l of the p-th cyclotomic polynomial.

{
p = 13;
forprime(l = 17, 101,
         U = Mod(1,l);
	 f = U*polcyclo(p);
         print(l, " ", lift(factor(f)))
	);
}
17 [x^6 + 5*x^5 + 2*x^4 + 4*x^3 + 2*x^2 + 5*x + 1, 1; x^6 + 13*x^5 + 2*x^4 + 12*x^3 + 2*x^2 + 13*x + 1, 1]
19 Mat([x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1])
23 [x^6 + 9*x^5 + 2*x^4 + 8*x^3 + 2*x^2 + 9*x + 1, 1; x^6 + 15*x^5 + 2*x^4 + 14*x^3 + 2*x^2 + 15*x + 1, 1]
29 [x^3 + 3*x^2 + 12*x + 28, 1; x^3 + 15*x^2 + 5*x + 28, 1; x^3 + 17*x^2 + 26*x + 28, 1; x^3 + 24*x^2 + 14*x + 28, 1]
31 [x^4 + 4*x^3 + 11*x^2 + 4*x + 1, 1; x^4 + 6*x^3 + 29*x^2 + 6*x + 1, 1; x^4 + 22*x^3 + 27*x^2 + 22*x + 1, 1]
37 Mat([x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1])
41 Mat([x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1])
43 [x^6 + 12*x^5 + 2*x^4 + 11*x^3 + 2*x^2 + 12*x + 1, 1; x^6 + 32*x^5 + 2*x^4 + 31*x^3 + 2*x^2 + 32*x + 1, 1]
47 [x^4 + 9*x^3 + 24*x^2 + 9*x + 1, 1; x^4 + 14*x^3 + 40*x^2 + 14*x + 1, 1; x^4 + 25*x^3 + 35*x^2 + 25*x + 1, 1]
53 [x + 4, 1; x + 6, 1; x + 7, 1; x + 9, 1; x + 11, 1; x + 17, 1; x + 25, 1; x + 29, 1; x + 37, 1; x + 38, 1; x + 40, 1; x + 43, 1]
59 Mat([x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1])
61 [x^3 + 15*x^2 + 38*x + 60, 1; x^3 + 23*x^2 + 46*x + 60, 1; x^3 + 34*x^2 + 10*x + 60, 1; x^3 + 51*x^2 + 27*x + 60, 1]
67 Mat([x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1])
71 Mat([x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1])
73 [x^4 + 20*x^3 + 14*x^2 + 20*x + 1, 1; x^4 + 61*x^3 + 9*x^2 + 61*x + 1, 1; x^4 + 66*x^3 + 55*x^2 + 66*x + 1, 1]
79 [x + 12, 1; x + 14, 1; x + 15, 1; x + 17, 1; x + 27, 1; x + 33, 1; x + 41, 1; x + 57, 1; x + 58, 1; x + 61, 1; x + 69, 1; x + 71, 1]
83 [x^4 + 8*x^3 + 55*x^2 + 8*x + 1, 1; x^4 + 30*x^3 + 39*x^2 + 30*x + 1, 1; x^4 + 46*x^3 + 77*x^2 + 46*x + 1, 1]
89 Mat([x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1])
97 Mat([x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1])
101 [x^6 + 18*x^5 + 2*x^4 + 17*x^3 + 2*x^2 + 18*x + 1, 1; x^6 + 84*x^5 + 2*x^4 + 83*x^3 + 2*x^2 + 84*x + 1, 1]