----------------------------------------------------------------------
| Sage Version 5.3, Release Date: 2012-09-08                         |
| Type "notebook()" for the browser-based notebook interface.        |
| Type "help()" for help.                                            |
----------------------------------------------------------------------
sage: F=QQ[2^(1/2)]
sage: F
Number Field in sqrt2 with defining polynomial x^2 - 2
sage: F.class_number()
1
sage: a=F.gen()
sage: a^2
2
sage: F.real_embeddings()
[
Ring morphism:
  From: Number Field in sqrt2 with defining polynomial x^2 - 2
  To:   Real Field with 53 bits of precision
  Defn: sqrt2 |--> -1.41421356237310,
Ring morphism:
  From: Number Field in sqrt2 with defining polynomial x^2 - 2
  To:   Real Field with 53 bits of precision
  Defn: sqrt2 |--> 1.41421356237310
]
sage: F.complex_embeddings()
[
Ring morphism:
  From: Number Field in sqrt2 with defining polynomial x^2 - 2
  To:   Complex Field with 53 bits of precision
  Defn: sqrt2 |--> -1.41421356237310,
Ring morphism:
  From: Number Field in sqrt2 with defining polynomial x^2 - 2
  To:   Complex Field with 53 bits of precision
  Defn: sqrt2 |--> 1.41421356237310
]
sage: E = QQ[(1+2^(1/3))^(1/2)]
sage: E
Number Field in a with defining polynomial x^6 - 3*x^4 + 3*x^2 - 3
sage: a = E.gen()
sage: a.minpoly()
x^6 - 3*x^4 + 3*x^2 - 3
sage: (a^2+1).minpoly()
x^3 - 6*x^2 + 12*x - 10
sage: E.real_embeddings()
[
Ring morphism:
  From: Number Field in a with defining polynomial x^6 - 3*x^4 + 3*x^2 - 3
  To:   Real Field with 53 bits of precision
  Defn: a |--> -1.50330337919359,
Ring morphism:
  From: Number Field in a with defining polynomial x^6 - 3*x^4 + 3*x^2 - 3
  To:   Real Field with 53 bits of precision
  Defn: a |--> 1.50330337919359
]
sage: E.complex_embeddings()
[
Ring morphism:
  From: Number Field in a with defining polynomial x^6 - 3*x^4 + 3*x^2 - 3
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -1.50330337919359,
Ring morphism:
  From: Number Field in a with defining polynomial x^6 - 3*x^4 + 3*x^2 - 3
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.872411215780490 - 0.625349385837253*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^6 - 3*x^4 + 3*x^2 - 3
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.872411215780490 + 0.625349385837253*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^6 - 3*x^4 + 3*x^2 - 3
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 0.872411215780490 - 0.625349385837253*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^6 - 3*x^4 + 3*x^2 - 3
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 0.872411215780490 + 0.625349385837253*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^6 - 3*x^4 + 3*x^2 - 3
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 1.50330337919359
]
sage: E.class_number()
1
sage: F.ring_of_integers()
Maximal Order in Number Field in sqrt2 with defining polynomial x^2 - 2
sage: O_F = F.ring_of_integers()
sage: O_E = E.ring_of_integers()
sage: O_F.gens()
[1, sqrt2]
sage: O_E.gens()
[1, a, a^2, a^3, a^4, a^5]
sage: E(2).factor() # E(2) means interpret 2 as an element of E, then factor (since UFD!)
(-3*a^5 + 9*a^4 - 10*a^3 + 6*a^2 - 3*a + 2) * (a + 1)^6
sage: