sage: 
sage: sage: 

sage: K = QQ[sqrt(2)]
sage: K.galois_group()
Galois group of Number Field in sqrt2 with defining polynomial x^2 - 2
sage: G=K.galois_group()
sage: G.order()
2
sage: K = QQ[sqrt(2+sqrt(2))]
sage: a = K.gen()
sage: G = K.galois_group()
sage: G.order()
4
sage: G.gens()
[(1,2,4,3)]
sage: s = G.gens()[0]
sage: s(a)
a^3 - 3*a
sage: # a^3-3a = sqrt(2-sqrt(2))
sage: s(a^3-3*a)
-a
sage: # this proves the statement from lecture 12
sage: 
sage: K = CyclotomicField(23)
sage: G = K.galois_group()
sage: G.gens()
[(1,8,10,7,12,17,15,21,6,20,5,14,3,2,13,19,22,18,9,4,11,16)]
sage: # thus cyclic since (Z/ 23)^x = Z/ 22 Z
sage: 
sage: K = QQ[2^(1/3)]
sage: K.is_galois()
False
sage: L = K.galois_closure('b')
sage: L.is_isomorphic(QQ[2^(1/3),(-1+sqrt(-3))/2])
True
sage: a = K.gen()
sage: b = L.gen()
sage: G = L.galois_group()
sage: G
Galois group of Number Field in b with defining polynomial x^6 + 40*x^3 + 1372
sage: G.gens()
[(1,2)(3,4)(5,6), (1,4,6)(2,5,3)]
sage: s,t = G.gens()
sage: s
(1,2)(3,4)(5,6)
sage: t
(1,4,6)(2,5,3)

sage: L.subfields()
[
(Number Field in b0 with defining polynomial x, Ring morphism:
  From: Number Field in b0 with defining polynomial x
  To:   Number Field in b with defining polynomial x^6 + 40*x^3 + 1372
  Defn: 0 |--> 0, None),
(Number Field in b1 with defining polynomial x^2 + 120*x + 12348, Ring morphism:
  From: Number Field in b1 with defining polynomial x^2 + 120*x + 12348
  To:   Number Field in b with defining polynomial x^6 + 40*x^3 + 1372
  Defn: b1 |--> 3*b^3, None),
(Number Field in b2 with defining polynomial x^3 - 128, Ring morphism:
  From: Number Field in b2 with defining polynomial x^3 - 128
  To:   Number Field in b with defining polynomial x^6 + 40*x^3 + 1372
  Defn: b2 |--> -2/63*b^4 + 32/63*b, None),
(Number Field in b3 with defining polynomial x^3 + 250, Ring morphism:
  From: Number Field in b3 with defining polynomial x^3 + 250
  To:   Number Field in b with defining polynomial x^6 + 40*x^3 + 1372
  Defn: b3 |--> 5/252*b^4 + 275/126*b, None),
(Number Field in b4 with defining polynomial x^3 - 2, Ring morphism:
  From: Number Field in b4 with defining polynomial x^3 - 2
  To:   Number Field in b with defining polynomial x^6 + 40*x^3 + 1372
  Defn: b4 |--> 1/84*b^4 + 13/42*b, None),
(Number Field in b5 with defining polynomial x^6 + 40*x^3 + 1372, Ring morphism:
  From: Number Field in b5 with defining polynomial x^6 + 40*x^3 + 1372
  To:   Number Field in b with defining polynomial x^6 + 40*x^3 + 1372
  Defn: b5 |--> b, Ring morphism:
  From: Number Field in b with defining polynomial x^6 + 40*x^3 + 1372
  To:   Number Field in b5 with defining polynomial x^6 + 40*x^3 + 1372
  Defn: b |--> b5)
]
sage: G.subgroups()
[Permutation Group with generators [()],
 Permutation Group with generators [(1,2)(3,4)(5,6)],
 Permutation Group with generators [(1,3)(2,6)(4,5)],
 Permutation Group with generators [(1,5)(2,4)(3,6)],
 Permutation Group with generators [(1,6,4)(2,3,5)],
 Permutation Group with generators [(1,2)(3,4)(5,6), (1,6,4)(2,3,5)]]
sage: s
(1,2)(3,4)(5,6)
sage: t
(1,4,6)(2,5,3)
sage: