┌────────────────────────────────────────────────────────────────────┐
│ Sage Version 6.0, Release Date: 2013-12-17                         │
│ Type "notebook()" for the browser-based notebook interface.        │
│ Type "help()" for help.                                            │
└────────────────────────────────────────────────────────────────────┘
sage: 


sage: K = CyclotomicField(23)
sage: L = QQ[(1+2^(1/3))^(1/2)]
sage: z = K.gen()
sage: a = L.gen()
sage: a
a
sage: a.minpoly()
x^6 - 3*x^4 + 3*x^2 - 3
sage: F = a.minpoly()
sage: F.base_ring()
Rational Field
sage: ZZ[x](F)
x^6 - 3*x^4 + 3*x^2 - 3
sage: GF(97)[x](F)
x^6 + 94*x^4 + 3*x^2 + 94
sage: GF(97)[x](F).factor()
(x^3 + 18*x^2 + 15*x + 87) * (x^3 + 79*x^2 + 15*x + 10)
sage: L.factor(97)
(Fractional ideal (97, a^3 + 18*a^2 + 15*a - 10)) * (Fractional ideal (97, a^3 - 18*a^2 + 15*a + 10))
sage: list(L.factor(97))
[(Fractional ideal (97, a^3 + 18*a^2 + 15*a - 10), 1),
 (Fractional ideal (97, a^3 - 18*a^2 + 15*a + 10), 1)]
sage: L.ring_of_integers().gens()
[1, a, a^2, a^3, a^4, a^5]
sage: T = QQ[19^(1/3)]
sage: a = T.gen()
sage: T.ring_of_integers().gens()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]
sage: T.factor(3)
(Fractional ideal (3, 1/3*a^2 + 1/3*a + 1/3))^2 * (Fractional ideal (3, 1/3*a^2 + 1/3*a - 2/3))
sage: list(T.factor(3))
[(Fractional ideal (3, 1/3*a^2 + 1/3*a + 1/3), 2),
 (Fractional ideal (3, 1/3*a^2 + 1/3*a - 2/3), 1)]
sage: GF(3)[x](a.minpoly()).factor()
(x + 2)^3
sage: