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│ Sage Version 6.0, Release Date: 2013-12-17                         │
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sage: 

sage: K1 = QQ[sqrt(5), sqrt(17)]
sage: K2 = CyclotomicField(19)
sage: K3 = QQ[7^(1/9)]
sage: K4 = QQ[sqrt(3), sqrt(5)]
sage: K4.units()
[1/2*sqrt5 + 1/2,
 (-1/2*sqrt5 - 1/2)*sqrt3 + 1/2*sqrt5 + 3/2,
 (-1/2*sqrt5 + 1/2)*sqrt3 - 1/2*sqrt5 + 3/2]
sage: K4.roots_of_unity()
[-1, 1]
sage: K4.unit_group()
Unit group with structure C2 x Z x Z x Z of Number Field in sqrt3 with defining polynomial x^2 - 3 over its base field
sage: K4.unit_group().gens()
(u0, u1, u2, u3)
sage: K1.units()
[1/2*sqrt5 - 1/2,
 (1/4*sqrt17 + 5/4)*sqrt5 - 3/4*sqrt17 - 11/4,
 (-1/4*sqrt17 + 3/4)*sqrt5 - 1/4*sqrt17 + 7/4]
sage: K1.roots_of_unity()
[-1, 1]
sage: K3
Number Field in a with defining polynomial x^9 - 7
sage: K3.units()
[a^3 - 2,
 2*a^8 - 2*a^7 + 2*a^6 - 2*a^5 + a^4 - 3*a^2 + 7*a - 10,
 a^8 + a^6 - 3*a^4 - a^3 - a^2 - 3*a + 5,
 6*a^8 - 3*a^7 + 9*a^6 - 30*a^5 + 33*a^4 - 18*a^3 + 30*a^2 - 45*a + 1]
sage: K3.roots_of_unity()
[-1, 1]
sage: K2
Cyclotomic Field of order 19 and degree 18
sage: K2.units()
[zeta19^17 + zeta19^5,
 zeta19^16 + zeta19^7 + zeta19^2,
 zeta19^9 + zeta19^5,
 zeta19^14 + zeta19^9,
 zeta19^9 + zeta19^6 + zeta19^3 + 1,
 zeta19^14 + zeta19,
 zeta19^15 + zeta19^14 + zeta19^13 + zeta19^12 + zeta19^11 + zeta19^10 + zeta19^9 + zeta19^8 + zeta19^7 + zeta19^6 + zeta19^5 + zeta19^4 + zeta19^3 + zeta19^2,
 zeta19^13 + zeta19^11]
sage: K2.roots_of_unity()
[1,
 zeta19,
 zeta19^2,
 zeta19^3,
 zeta19^4,
 zeta19^5,
 zeta19^6,
 zeta19^7,
 zeta19^8,
 zeta19^9,
 zeta19^10,
 zeta19^11,
 zeta19^12,
 zeta19^13,
 zeta19^14,
 zeta19^15,
 zeta19^16,
 zeta19^17,
 -zeta19^17 - zeta19^16 - zeta19^15 - zeta19^14 - zeta19^13 - zeta19^12 - zeta19^11 - zeta19^10 - zeta19^9 - zeta19^8 - zeta19^7 - zeta19^6 - zeta19^5 - zeta19^4 - zeta19^3 - zeta19^2 - zeta19 - 1,
 -1,
 -zeta19,
 -zeta19^2,
 -zeta19^3,
 -zeta19^4,
 -zeta19^5,
 -zeta19^6,
 -zeta19^7,
 -zeta19^8,
 -zeta19^9,
 -zeta19^10,
 -zeta19^11,
 -zeta19^12,
 -zeta19^13,
 -zeta19^14,
 -zeta19^15,
 -zeta19^16,
 -zeta19^17,
 zeta19^17 + zeta19^16 + zeta19^15 + zeta19^14 + zeta19^13 + zeta19^12 + zeta19^11 + zeta19^10 + zeta19^9 + zeta19^8 + zeta19^7 + zeta19^6 + zeta19^5 + zeta19^4 + zeta19^3 + zeta19^2 + zeta19 + 1]
sage: