CoCalc -- 1721.sagews

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x = PolynomialRing(QQ, 'x').gen()
K.<a>=CyclotomicField(17)
K

Cyclotomic Field of order 17 and degree 16

K.unit_group().gens()

[-a^3, a^3 + a^2, a^14 + a^3, a^12 + a^11 + a^10 + a^9 + a^8 + a^7 + a^6 + a^5 + a^4 + a^3 + a^2, a^15 + a, a^8 + a^3, a^10 + a^8, a^15 + a^6]

K.unit_group()

Unit group with structure C34 x Z x Z x Z x Z x Z x Z x Z of Cyclotomic Field of order 17 and degree 16

L.<b>=CyclotomicField(68)

K.embeddings(L)

[
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> -b^2,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> b^4,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> -b^6,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> b^8,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> -b^10,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> b^12,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> -b^14,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> b^16,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> -b^18,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> b^20,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> -b^22,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> b^24,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> -b^26,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> b^28,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> b^30 - b^28 + b^26 - b^24 + b^22 - b^20 + b^18 - b^16 + b^14 - b^12 + b^10 - b^8 + b^6 - b^4 + b^2 - 1,
Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: a |--> -b^30
]

L.subfields()

[
(Number Field in b0 with defining polynomial x, Ring morphism:
  From: Number Field in b0 with defining polynomial x
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: 0 |--> 0, None),
(Number Field in b1 with defining polynomial x^2 + 1, Ring morphism:
  From: Number Field in b1 with defining polynomial x^2 + 1
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b1 |--> -b^17, None),
(Number Field in b2 with defining polynomial x^2 + 17, Ring morphism:
  From: Number Field in b2 with defining polynomial x^2 + 17
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b2 |--> 2*b^31 - 2*b^29 + 2*b^27 + 2*b^23 - b^17 + 2*b^11 + 2*b^7 - 2*b^5 + 2*b^3, None),
(Number Field in b3 with defining polynomial x^2 - 2*x - 16, Ring morphism:
  From: Number Field in b3 with defining polynomial x^2 - 2*x - 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b3 |--> 2*b^28 + 2*b^24 - 2*b^22 + 2*b^20 - 2*b^14 + 2*b^12 - 2*b^10 - 2*b^6 + 2, None),
(Number Field in b4 with defining polynomial x^4 + 9*x^2 + 16, Ring morphism:
  From: Number Field in b4 with defining polynomial x^4 + 9*x^2 + 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b4 |--> b^31 - b^29 + b^27 + b^23 - b^17 + b^11 + b^7 - b^5 + b^3, None),
(Number Field in b5 with defining polynomial x^4 + 17*x^2 + 68, Ring morphism:
  From: Number Field in b5 with defining polynomial x^4 + 17*x^2 + 68
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b5 |--> b^31 - b^29 + b^27 - 2*b^25 + b^23 + 2*b^19 - b^17 + 2*b^15 + b^11 - 2*b^9 + b^7 - b^5 + b^3, None),
(Number Field in b6 with defining polynomial x^4 - 2*x^3 - 24*x^2 + 8*x + 16, Ring morphism:
  From: Number Field in b6 with defining polynomial x^4 - 2*x^3 - 24*x^2 + 8*x + 16
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b6 |--> -2*b^30 + 2*b^28 + 2*b^24 - 2*b^22 + 2*b^20 - 2*b^18 + 2*b^16 - 2*b^14 + 2*b^12 - 2*b^10 - 2*b^6 + 2*b^4 + 2, None),
(Number Field in b7 with defining polynomial x^8 + 13*x^6 + 40*x^4 + 13*x^2 + 1, Ring morphism:
  From: Number Field in b7 with defining polynomial x^8 + 13*x^6 + 40*x^4 + 13*x^2 + 1
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b7 |--> b^31 - b^29 + b^27 - b^25 + b^23 + b^19 - b^17 + b^15 + b^11 - b^9 + b^7 - b^5 + b^3, None),
(Number Field in b8 with defining polynomial x^8 + 17*x^6 + 68*x^4 + 85*x^2 + 17, Ring morphism:
  From: Number Field in b8 with defining polynomial x^8 + 17*x^6 + 68*x^4 + 85*x^2 + 17
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b8 |--> b^31 - b^29 + b^27 - b^25 + b^23 - 2*b^21 + b^19 - b^17 + b^15 - 2*b^13 + b^11 - b^9 + b^7 - b^5 + b^3, None),
(Number Field in b9 with defining polynomial x^8 - 2*x^7 - 28*x^6 + 48*x^5 + 240*x^4 - 320*x^3 - 640*x^2 + 512*x + 256, Ring morphism:
  From: Number Field in b9 with defining polynomial x^8 - 2*x^7 - 28*x^6 + 48*x^5 + 240*x^4 - 320*x^3 - 640*x^2 + 512*x + 256
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b9 |--> -2*b^30 + 2*b^28 - 2*b^26 + 2*b^24 - 2*b^22 + 2*b^20 - 2*b^18 + 2*b^16 - 2*b^14 + 2*b^12 - 2*b^10 + 2*b^8 - 2*b^6 + 2*b^4 + 2, None),
(Number Field in b10 with defining polynomial x^16 + 15*x^14 + 91*x^12 + 286*x^10 + 495*x^8 + 462*x^6 + 210*x^4 + 36*x^2 + 1, Ring morphism:
  From: Number Field in b10 with defining polynomial x^16 + 15*x^14 + 91*x^12 + 286*x^10 + 495*x^8 + 462*x^6 + 210*x^4 + 36*x^2 + 1
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b10 |--> b^31 - b^29 + b^27 - b^25 + b^23 - b^21 + b^19 - b^17 + b^15 - b^13 + b^11 - b^9 + b^7 - b^5 + b^3, None),
(Number Field in b11 with defining polynomial x^16 - 17*x^14 + 119*x^12 - 442*x^10 + 935*x^8 - 1122*x^6 + 714*x^4 - 204*x^2 + 17, Ring morphism:
  From: Number Field in b11 with defining polynomial x^16 - 17*x^14 + 119*x^12 - 442*x^10 + 935*x^8 - 1122*x^6 + 714*x^4 - 204*x^2 + 17
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b11 |--> -b^31 + b^29 - b^27 + b^25 - b^23 + b^21 - b^19 + b^17 - b^15 + b^13 - b^11 + b^9 - b^7 + b^5 - b^3 + 2*b, None),
(Number Field in b12 with defining polynomial x^16 - 2*x^15 + 4*x^14 - 8*x^13 + 16*x^12 - 32*x^11 + 64*x^10 - 128*x^9 + 256*x^8 - 512*x^7 + 1024*x^6 - 2048*x^5 + 4096*x^4 - 8192*x^3 + 16384*x^2 - 32768*x + 65536, Ring morphism:
  From: Number Field in b12 with defining polynomial x^16 - 2*x^15 + 4*x^14 - 8*x^13 + 16*x^12 - 32*x^11 + 64*x^10 - 128*x^9 + 256*x^8 - 512*x^7 + 1024*x^6 - 2048*x^5 + 4096*x^4 - 8192*x^3 + 16384*x^2 - 32768*x + 65536
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b12 |--> 2*b^2, None),
(Number Field in b13 with defining polynomial x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1, Ring morphism:
  From: Number Field in b13 with defining polynomial x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
  To:   Cyclotomic Field of order 68 and degree 32
  Defn: b13 |--> b, Ring morphism:
  From: Cyclotomic Field of order 68 and degree 32
  To:   Number Field in b13 with defining polynomial x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
  Defn: b |--> b13)
]

K.subfields()

[
(Number Field in a0 with defining polynomial x + 1, Ring morphism:
  From: Number Field in a0 with defining polynomial x + 1
  To:   Cyclotomic Field of order 17 and degree 16
  Defn: -1 |--> -1, None),
(Number Field in a1 with defining polynomial x^2 + x - 4, Ring morphism:
  From: Number Field in a1 with defining polynomial x^2 + x - 4
  To:   Cyclotomic Field of order 17 and degree 16
  Defn: a1 |--> -a^14 - a^12 - a^11 - a^10 - a^7 - a^6 - a^5 - a^3 - 1, None),
(Number Field in a2 with defining polynomial x^4 + x^3 - 6*x^2 - x + 1, Ring morphism:
  From: Number Field in a2 with defining polynomial x^4 + x^3 - 6*x^2 - x + 1
  To:   Cyclotomic Field of order 17 and degree 16
  Defn: a2 |--> -a^15 - a^14 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^3 - a^2 - 1, None),
(Number Field in a3 with defining polynomial x^8 + x^7 - 7*x^6 - 6*x^5 + 15*x^4 + 10*x^3 - 10*x^2 - 4*x + 1, Ring morphism:
  From: Number Field in a3 with defining polynomial x^8 + x^7 - 7*x^6 - 6*x^5 + 15*x^4 + 10*x^3 - 10*x^2 - 4*x + 1
  To:   Cyclotomic Field of order 17 and degree 16
  Defn: a3 |--> -a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 - 1, None),
(Number Field in a4 with defining polynomial x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, Ring morphism:
  From: Number Field in a4 with defining polynomial x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
  To:   Cyclotomic Field of order 17 and degree 16
  Defn: a4 |--> a, Ring morphism:
  From: Cyclotomic Field of order 17 and degree 16
  To:   Number Field in a4 with defining polynomial x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
  Defn: a |--> a4)
]

F.<t>=NumberField(x^4 + x^3 - 6*x^2 - x + 1)

F.unit_group()

Unit group with structure C2 x Z x Z x Z of Number Field in t with defining polynomial x^4 + x^3 - 6*x^2 - x + 1

F.unit_group().gens()

[-1, 1/2*t^3 + t^2 - 2*t - 3/2, t, 1/2*t^3 - 4*t + 3/2]

minpoly(1/2*t^3 + t^2 - 2*t - 3/2)

x^4 - x^3 - 6*x^2 + x + 1

minpoly(1/2*t^3 - 4*t + 3/2)

x^4 - 2*x^3 - 7*x^2 + 8*x - 1

minpoly(t^2)

x^4 - 13*x^3 + 40*x^2 - 13*x + 1

E.<s>=NumberField(x^4 + 9*x^2 + 16)
E

Number Field in s with defining polynomial x^4 + 9*x^2 + 16

E.unit_group()

Unit group with structure C4 x Z of Number Field in s with defining polynomial x^4 + 9*x^2 + 16

K.factor(-11)

(Fractional ideal (a^8 - a))^10

b1=a-a^10

b2=a^2-a^9

b3=a^3-a^8

b4=a^4-a^7

b5=a^5-a^6

(b1-b2+b3+b4+b5)^2

-11

minpoly(b1)

x^10 + 11*x^8 + 44*x^6 + 77*x^4 + 55*x^2 + 11

minpoly(b2)

x^10 + 11*x^8 + 44*x^6 + 77*x^4 + 55*x^2 + 11

minpoly(b1-b2+b3+b4+b5)

x^2 + 11

minpoly(a+a^10)

x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1

((a+a^4)*a).minpoly().disc().factor()

5^3