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

sage: K=QQ[5^(1/2)]
sage: K.ring_of_integers()
Maximal Order in Number Field in sqrt5 with defining polynomial x^2 - 5
sage: O_K=K.ring_of_integers()
sage: O_K.gens()
[1/2*sqrt5 + 1/2, sqrt5]
sage: O_K.is_integrally_closed()
True
sage: ZZ[5^(1/2)].is_integrally_closed()
False
sage: ZZ[5^(1/2)].integral_closure()
Maximal Order in Number Field in sqrt5 with defining polynomial x^2 - 5
sage: ZZ[5^(1/2)].integral_closure().gens()
[1/2*sqrt5 + 1/2, sqrt5]
sage: QQ[3^(1/2)].ring_of_integers().gens()
[1, sqrt3]
sage: 
sage: F=QQ[(1+2^(1/3))^(1/2)]
sage: F.ring_of_integers().gens()
[1, a, a^2, a^3, a^4, a^5]
sage: 
sage: QQ[exp(2*pi*I/17)].ring_of_integers().gens()
[1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15]
sage: