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│ Sage Version 6.0, Release Date: 2013-12-17                         │
│ Type "notebook()" for the browser-based notebook interface.        │
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sage: 
sage: 
sage: n = 3*5*7
sage: n
105
sage: K = CyclotomicField(n)
sage: K.degree()
48
sage: p = 3
sage: K.factor(p)
(Fractional ideal (3, -3*zeta105^40 + zeta105^12 - zeta105^11 - zeta105^10 + zeta105^9 - zeta105^8 + zeta105^7 - 2*zeta105^5 - zeta105^4 + zeta105^2 + 1))^2 * (Fractional ideal (3, -3*zeta105^40 + zeta105^12 + zeta105^10 - zeta105^8 + zeta105^7 - 2*zeta105^5 - zeta105^4 + zeta105^3 - zeta105^2 + 2*zeta105 + 1))^2
sage: list(K.factor(p))
[(Fractional ideal (3, -3*zeta105^40 + zeta105^12 - zeta105^11 - zeta105^10 + zeta105^9 - zeta105^8 + zeta105^7 - 2*zeta105^5 - zeta105^4 + zeta105^2 + 1),
  2),
 (Fractional ideal (3, -3*zeta105^40 + zeta105^12 + zeta105^10 - zeta105^8 + zeta105^7 - 2*zeta105^5 - zeta105^4 + zeta105^3 - zeta105^2 + 2*zeta105 + 1),
  2)]
sage: I = K.factor(p)[0][0]
sage: I
Fractional ideal (3, -3*zeta105^40 + zeta105^12 - zeta105^11 - zeta105^10 + zeta105^9 - zeta105^8 + zeta105^7 - 2*zeta105^5 - zeta105^4 + zeta105^2 + 1)
sage: I.residue_field()
Residue field in zeta105bar of Fractional ideal (3, -3*zeta105^40 + zeta105^12 - zeta105^11 - zeta105^10 + zeta105^9 - zeta105^8 + zeta105^7 - 2*zeta105^5 - zeta105^4 + zeta105^2 + 1)
sage: I.residue_field().cardinality()
531441
sage: 531441.factor()
3^12
sage: f_p1_p = 12
sage: e_p1_p = 2
sage: J = K.factor(p)[1][0]

sage: J.residue_field().cardinality().factor()
3^12
sage: 12*2+12*2==48
True
sage: log(3^12)
log(531441)
sage: log(3^12).n()
13.1833474640173
sage: log(3^12,3)
12
sage: sum(log(I[0].residue_field().cardinality(), p)* I[1] for I in K.factor(p))
48
sage: p = 5
sage: sum(log(I[0].residue_field().cardinality(), p)* I[1] for I in K.factor(p))
48
sage: K.factor(2)
(Fractional ideal (2, zeta105^12 + zeta105^10 + zeta105^9 + zeta105^7 + zeta105^6 + zeta105^4 + 1)) * (Fractional ideal (2, zeta105^12 + zeta105^11 + zeta105^9 + zeta105^8 + zeta105^7 + zeta105^3 + 1)) * (Fractional ideal (2, -2*zeta105^40 + zeta105^12 + zeta105^9 - zeta105^5 + zeta105^4 + zeta105^3 + zeta105 + 1)) * (Fractional ideal (2, zeta105^12 + zeta105^8 + zeta105^6 + zeta105^5 - zeta105^3 + zeta105^2 + 1))
sage: K.discriminant().factor()
3^24 * 5^36 * 7^40
sage: L = QQ[2^(1/3)]
sage: L.degree()
3
sage: L.factor(2)
(Fractional ideal (a))^3
sage: L.discriminant()
-108
sage: 108.factor(
....: )
2^2 * 3^3
sage: L.factor(3)
(Fractional ideal (a + 1))^3
sage: