CoCalc -- lecture2.sagews

Number fields and embeddings

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Path: Algebraic number theory Spring 2018/lecture2.sagews

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K1 = QQ[sqrt(2)]
K2 = QQ[sqrt(2),sqrt(3)]

K1; K1.absolute_degree(); K1.gens(); K1.primitive_element();
K2; K2.absolute_degree(); K2.gens(); K2.primitive_element();

Number Field in sqrt2 with defining polynomial x^2 - 2
2
(sqrt2,)
sqrt2
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
4
(sqrt2, sqrt3)
sqrt2 - sqrt3

# K3 is the splitting field of x^3-2 equal to Q[zeta_3, cube root of 2]

K3 = QQ.extension([x^3-2,x^2+x+1],'a,z')
K3; K3.absolute_degree(); K3.gens(); K3.primitive_element()

K3a = QQ.extension(x^3-2,'a')
K3a; K3a.absolute_degree(); K3a.gens(); K3a.primitive_element()

a, z = K3.gens()

"a^3 = ", a^3; "z^2+z+1 = ", z^2+z+1

Number Field in a with defining polynomial x^3 - 2 over its base field
6
(a, z)
a - z
Number Field in a with defining polynomial x^3 - 2
3
(a,)
a
('a^3 = ', 2)
('z^2+z+1 = ', 0)


print "QQ(2^1/2) (degree %d) into CC"%(K1.absolute_degree())
K1.embeddings(CC)

print "QQ(2^1/2, 3^1/2) (degree %d) into CC"%(K2.absolute_degree())
K2.embeddings(CC)

print "QQ(2^1/3) (degree %d) into CC"%(K3a.absolute_degree())
K3a.embeddings(CC)
print "Embeddings into RR"
K3a.embeddings(RR)

print "QQ(2^1/3, zeta_3) (degree %d) into CC"%(K3.absolute_degree())
K3.embeddings(CC)

QQ(2^1/2) (degree 2) into CC
[
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
]
QQ(2^1/2, 3^1/2) (degree 4) into CC
[
Relative number field morphism:
  From: Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: sqrt2 |--> -1.41421356237310
        sqrt3 |--> 1.73205080756888,
Relative number field morphism:
  From: Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: sqrt2 |--> 1.41421356237310
        sqrt3 |--> 1.73205080756888,
Relative number field morphism:
  From: Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: sqrt2 |--> -1.41421356237310
        sqrt3 |--> -1.73205080756888,
Relative number field morphism:
  From: Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: sqrt2 |--> 1.41421356237310
        sqrt3 |--> -1.73205080756888
]
QQ(2^1/3) (degree 3) into CC
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.629960524947437 - 1.09112363597172*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.629960524947437 + 1.09112363597172*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 1.25992104989487
]
Embeddings into RR
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Real Field with 53 bits of precision
  Defn: a |--> 1.25992104989487
]
QQ(2^1/3, zeta_3) (degree 6) into CC
[
Relative number field morphism:
  From: Number Field in a with defining polynomial x^3 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.629960524947434 - 1.09112363597172*I
        z |--> -0.499999999999998 + 0.866025403784439*I,
Relative number field morphism:
  From: Number Field in a with defining polynomial x^3 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.629960524947436 - 1.09112363597172*I
        z |--> -0.500000000000000 - 0.866025403784439*I,
Relative number field morphism:
  From: Number Field in a with defining polynomial x^3 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.629960524947436 + 1.09112363597172*I
        z |--> -0.500000000000000 + 0.866025403784439*I,
Relative number field morphism:
  From: Number Field in a with defining polynomial x^3 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.629960524947434 + 1.09112363597172*I
        z |--> -0.499999999999998 - 0.866025403784439*I,
Relative number field morphism:
  From: Number Field in a with defining polynomial x^3 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 1.25992104989487 + 2.22044604925031e-16*I
        z |--> -0.500000000000000 + 0.866025403784438*I,
Relative number field morphism:
  From: Number Field in a with defining polynomial x^3 - 2 over its base field
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 1.25992104989487 - 2.22044604925031e-16*I
        z |--> -0.500000000000000 - 0.866025403784438*I
]