Galois groups of number fields¶

AUTHORS:

William Stein (2004, 2005): initial version
David Loeffler (2009): rewrote to give explicit homomorphism groups

sage.rings.number_field.galois_group.GaloisGroup[source]¶: alias of GaloisGroup_v1

class sage.rings.number_field.galois_group.GaloisGroupElement[source]¶

Bases: PermutationGroupElement

An element of a Galois group. This is stored as a permutation, but may also be made to act on elements of the field (generally returning elements of its Galois closure).

EXAMPLES:

Sage

sage: K.<w> = QuadraticField(-7); G = K.galois_group()
sage: G[1]
(1,2)
sage: G[1](w + 2)
-w + 2

sage: x = polygen(ZZ, 'x')
sage: L.<v> = NumberField(x^3 - 2); G = L.galois_group(names='y')
sage: G[4]
(1,5)(2,4)(3,6)
sage: G[4](v)
1/18*y^4
sage: G[4](G[4](v))
-1/36*y^4 - 1/2*y
sage: G[4](G[4](G[4](v)))
1/18*y^4

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(7), names=('w',)); (w,) = K._first_ngens(1); G = K.galois_group()
>>> G[Integer(1)]
(1,2)
>>> G[Integer(1)](w + Integer(2))
-w + 2

>>> x = polygen(ZZ, 'x')
>>> L = NumberField(x**Integer(3) - Integer(2), names=('v',)); (v,) = L._first_ngens(1); G = L.galois_group(names='y')
>>> G[Integer(4)]
(1,5)(2,4)(3,6)
>>> G[Integer(4)](v)
1/18*y^4
>>> G[Integer(4)](G[Integer(4)](v))
-1/36*y^4 - 1/2*y
>>> G[Integer(4)](G[Integer(4)](G[Integer(4)](v)))
1/18*y^4

Sage Live

K.<w> = QuadraticField(-7); G = K.galois_group()
G[1]
G[1](w + 2)
x = polygen(ZZ, 'x')
L.<v> = NumberField(x^3 - 2); G = L.galois_group(names='y')
G[4]
G[4](v)
G[4](G[4](v))
G[4](G[4](G[4](v)))

as_hom()[source]¶

Return the homomorphism \(L \to L\) corresponding to self, where \(L\) is the Galois closure of the ambient number field.

EXAMPLES:

Sage

sage: G = QuadraticField(-7,'w').galois_group()
sage: G[1].as_hom()
Ring endomorphism of Number Field in w with defining polynomial x^2 + 7
 with w = 2.645751311064591?*I
  Defn: w |--> -w

Python

>>> from sage.all import *
>>> G = QuadraticField(-Integer(7),'w').galois_group()
>>> G[Integer(1)].as_hom()
Ring endomorphism of Number Field in w with defining polynomial x^2 + 7
 with w = 2.645751311064591?*I
  Defn: w |--> -w

Sage Live

G = QuadraticField(-7,'w').galois_group()
G[1].as_hom()

ramification_degree(P)[source]¶

Return the greatest value of \(v\) such that \(s\) acts trivially modulo \(P^v\). Should only be used if \(P\) is prime and \(s\) is in the decomposition group of \(P\).

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<b> = NumberField(x^3 - 3, 'a').galois_closure()
sage: G = K.galois_group()
sage: P = K.primes_above(3)[0]
sage: s = hom(K, K, 1/18*b^4 - 1/2*b)
sage: G(s).ramification_degree(P)
4

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(3), 'a').galois_closure(names=('b',)); (b,) = K._first_ngens(1)
>>> G = K.galois_group()
>>> P = K.primes_above(Integer(3))[Integer(0)]
>>> s = hom(K, K, Integer(1)/Integer(18)*b**Integer(4) - Integer(1)/Integer(2)*b)
>>> G(s).ramification_degree(P)
4

Sage Live

x = polygen(ZZ, 'x')
K.<b> = NumberField(x^3 - 3, 'a').galois_closure()
G = K.galois_group()
P = K.primes_above(3)[0]
s = hom(K, K, 1/18*b^4 - 1/2*b)
G(s).ramification_degree(P)

class sage.rings.number_field.galois_group.GaloisGroup_subgroup(ambient, gens=None, gap_group=None, domain=None, category=None, canonicalize=True, check=True)[source]¶

Bases: GaloisSubgroup_perm

A subgroup of a Galois group, as returned by functions such as decomposition_group.

INPUT:

ambient – the ambient Galois group
gens – list of generators for the group
gap_group – a gap or libgap permutation group, or a string
defining one (default: None)
domain – set on which this permutation group acts; extracted from ambient if not specified
category – the category for this object
canonicalize – if True, sorts and removes duplicates
check – whether to check that generators actually lie in the ambient group

EXAMPLES:

Sage

sage: from sage.rings.number_field.galois_group import GaloisGroup_subgroup
    sage: x = polygen(ZZ, 'x')
sage: G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group()
sage: GaloisGroup_subgroup( G, [G([(1,2,3),(4,5,6)])])
Subgroup generated by [(1,2,3)(4,5,6)] of
 (Galois group 6T2 ([3]2) with order 6 of x^6 - 6*x^4 + 9*x^2 + 23)

sage: K.<a> = NumberField(x^6 - 3*x^2 - 1)
sage: L.<b> = K.galois_closure()
sage: G = L.galois_group()
sage: P = L.primes_above(3)[0]
sage: H = G.decomposition_group(P)
sage: H.order()
3

sage: G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group()
sage: H = G.subgroup([G([(1,2,3),(4,5,6)])])
sage: H
Subgroup generated by [(1,2,3)(4,5,6)] of
 (Galois group 6T2 ([3]2) with order 6 of x^6 - 6*x^4 + 9*x^2 + 23)

Python

>>> from sage.all import *
>>> from sage.rings.number_field.galois_group import GaloisGroup_subgroup
>>> x = polygen(ZZ, 'x')
>>> G = NumberField(x**Integer(3) - x - Integer(1), 'a').galois_closure('b').galois_group()
>>> GaloisGroup_subgroup( G, [G([(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5),Integer(6))])])
Subgroup generated by [(1,2,3)(4,5,6)] of
 (Galois group 6T2 ([3]2) with order 6 of x^6 - 6*x^4 + 9*x^2 + 23)

>>> K = NumberField(x**Integer(6) - Integer(3)*x**Integer(2) - Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.galois_closure(names=('b',)); (b,) = L._first_ngens(1)
>>> G = L.galois_group()
>>> P = L.primes_above(Integer(3))[Integer(0)]
>>> H = G.decomposition_group(P)
>>> H.order()
3

>>> G = NumberField(x**Integer(3) - x - Integer(1), 'a').galois_closure('b').galois_group()
>>> H = G.subgroup([G([(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5),Integer(6))])])
>>> H
Subgroup generated by [(1,2,3)(4,5,6)] of
 (Galois group 6T2 ([3]2) with order 6 of x^6 - 6*x^4 + 9*x^2 + 23)

Sage Live

from sage.rings.number_field.galois_group import GaloisGroup_subgroup
x = polygen(ZZ, 'x')
G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group()
GaloisGroup_subgroup( G, [G([(1,2,3),(4,5,6)])])
K.<a> = NumberField(x^6 - 3*x^2 - 1)
L.<b> = K.galois_closure()
G = L.galois_group()
P = L.primes_above(3)[0]
H = G.decomposition_group(P)
H.order()
G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group()
H = G.subgroup([G([(1,2,3),(4,5,6)])])
H

Element[source]¶: alias of GaloisGroupElement

fixed_field(name=None, polred=None, threshold=None)[source]¶

Return the fixed field of this subgroup (as a subfield of the Galois closure of the number field associated to the ambient Galois group).

INPUT:

name – a variable name for the new field
polred – whether to optimize the generator of the newly created field
for a simpler polynomial, using PARI’s pari:polredbest. Defaults to True when the degree of the fixed field is at most 8.
threshold – positive number; polred only performed if the cost is at most this threshold

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: L.<a> = NumberField(x^4 + 1)
sage: G = L.galois_group()
sage: H = G.decomposition_group(L.primes_above(3)[0])
sage: H.fixed_field()
(Number Field in a0 with defining polynomial x^2 + 2 with a0 = a^3 + a,
 Ring morphism:
   From: Number Field in a0 with defining polynomial x^2 + 2 with a0 = a^3 + a
   To:   Number Field in a with defining polynomial x^4 + 1
   Defn: a0 |--> a^3 + a)

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> L = NumberField(x**Integer(4) + Integer(1), names=('a',)); (a,) = L._first_ngens(1)
>>> G = L.galois_group()
>>> H = G.decomposition_group(L.primes_above(Integer(3))[Integer(0)])
>>> H.fixed_field()
(Number Field in a0 with defining polynomial x^2 + 2 with a0 = a^3 + a,
 Ring morphism:
   From: Number Field in a0 with defining polynomial x^2 + 2 with a0 = a^3 + a
   To:   Number Field in a with defining polynomial x^4 + 1
   Defn: a0 |--> a^3 + a)

Sage Live

x = polygen(ZZ, 'x')
L.<a> = NumberField(x^4 + 1)
G = L.galois_group()
H = G.decomposition_group(L.primes_above(3)[0])
H.fixed_field()

You can use the polred option to get a simpler defining polynomial:

Sage

sage: K.<a> = NumberField(x^5 - 5*x^2 - 3)
sage: G = K.galois_group(); G
Galois group 5T2 (5:2) with order 10 of x^5 - 5*x^2 - 3
sage: sigma, tau = G.gens()
sage: H = G.subgroup([tau])
sage: H.fixed_field(polred=False)
(Number Field in a0 with defining polynomial x^2 + 84375
  with a0 = 5*ac^5 + 25*ac^3,
 Ring morphism:
   From: Number Field in a0 with defining polynomial x^2 + 84375
         with a0 = 5*ac^5 + 25*ac^3
   To:   Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375
   Defn: a0 |--> 5*ac^5 + 25*ac^3)
sage: H.fixed_field(polred=True)
(Number Field in a0 with defining polynomial x^2 - x + 4
  with a0 = -1/30*ac^5 - 1/6*ac^3 + 1/2,
 Ring morphism:
   From: Number Field in a0 with defining polynomial x^2 - x + 4
         with a0 = -1/30*ac^5 - 1/6*ac^3 + 1/2
   To:   Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375
   Defn: a0 |--> -1/30*ac^5 - 1/6*ac^3 + 1/2)
sage: G.splitting_field()
Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(5) - Integer(5)*x**Integer(2) - Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> G = K.galois_group(); G
Galois group 5T2 (5:2) with order 10 of x^5 - 5*x^2 - 3
>>> sigma, tau = G.gens()
>>> H = G.subgroup([tau])
>>> H.fixed_field(polred=False)
(Number Field in a0 with defining polynomial x^2 + 84375
  with a0 = 5*ac^5 + 25*ac^3,
 Ring morphism:
   From: Number Field in a0 with defining polynomial x^2 + 84375
         with a0 = 5*ac^5 + 25*ac^3
   To:   Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375
   Defn: a0 |--> 5*ac^5 + 25*ac^3)
>>> H.fixed_field(polred=True)
(Number Field in a0 with defining polynomial x^2 - x + 4
  with a0 = -1/30*ac^5 - 1/6*ac^3 + 1/2,
 Ring morphism:
   From: Number Field in a0 with defining polynomial x^2 - x + 4
         with a0 = -1/30*ac^5 - 1/6*ac^3 + 1/2
   To:   Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375
   Defn: a0 |--> -1/30*ac^5 - 1/6*ac^3 + 1/2)
>>> G.splitting_field()
Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375

Sage Live

K.<a> = NumberField(x^5 - 5*x^2 - 3)
G = K.galois_group(); G
sigma, tau = G.gens()
H = G.subgroup([tau])
H.fixed_field(polred=False)
H.fixed_field(polred=True)
G.splitting_field()

An embedding is returned also if the subgroup is trivial (Issue #26817):

Sage

sage: H = G.subgroup([])
sage: H.fixed_field()
(Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375,
 Identity endomorphism of
  Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375)

Python

>>> from sage.all import *
>>> H = G.subgroup([])
>>> H.fixed_field()
(Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375,
 Identity endomorphism of
  Number Field in ac with defining polynomial x^10 + 10*x^8 + 25*x^6 + 3375)

Sage Live

H = G.subgroup([])
H.fixed_field()

class sage.rings.number_field.galois_group.GaloisGroup_v1(group, number_field)[source]¶

Bases: SageObject

A wrapper around a class representing an abstract transitive group.

This is just a fairly minimal object at present. To get the underlying group, do G.group(), and to get the corresponding number field do G.number_field(). For a more sophisticated interface use the type=None option.

EXAMPLES:

Sage

sage: # needs sage.symbolic
sage: from sage.rings.number_field.galois_group import GaloisGroup_v1
sage: K = QQ[2^(1/3)]
sage: pK = K.absolute_polynomial()
sage: G = GaloisGroup_v1(pK.galois_group(pari_group=True), K); G
...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
See https://github.com/sagemath/sage/issues/28782 for details.
Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the
 Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?
sage: G.order()
6
sage: G.group()
PARI group [6, -1, 2, "S3"] of degree 3
sage: G.number_field()
Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> from sage.rings.number_field.galois_group import GaloisGroup_v1
>>> K = QQ[Integer(2)**(Integer(1)/Integer(3))]
>>> pK = K.absolute_polynomial()
>>> G = GaloisGroup_v1(pK.galois_group(pari_group=True), K); G
...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
See https://github.com/sagemath/sage/issues/28782 for details.
Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the
 Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?
>>> G.order()
6
>>> G.group()
PARI group [6, -1, 2, "S3"] of degree 3
>>> G.number_field()
Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?

Sage Live

# needs sage.symbolic
from sage.rings.number_field.galois_group import GaloisGroup_v1
K = QQ[2^(1/3)]
pK = K.absolute_polynomial()
G = GaloisGroup_v1(pK.galois_group(pari_group=True), K); G
G.order()
G.group()
G.number_field()

group()[source]¶

Return the underlying abstract group.

EXAMPLES:

Sage

sage: from sage.rings.number_field.galois_group import GaloisGroup_v1
sage: x = polygen(ZZ, 'x')
sage: K = NumberField(x^3 + 2*x + 2, 'theta')
sage: G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K)
...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
See https://github.com/sagemath/sage/issues/28782 for details.
sage: H = G.group(); H
PARI group [6, -1, 2, "S3"] of degree 3
sage: P = H.permutation_group(); P
Transitive group number 2 of degree 3
sage: sorted(P)
[(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3)]

Python

>>> from sage.all import *
>>> from sage.rings.number_field.galois_group import GaloisGroup_v1
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(2)*x + Integer(2), 'theta')
>>> G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K)
...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
See https://github.com/sagemath/sage/issues/28782 for details.
>>> H = G.group(); H
PARI group [6, -1, 2, "S3"] of degree 3
>>> P = H.permutation_group(); P
Transitive group number 2 of degree 3
>>> sorted(P)
[(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3)]

Sage Live

from sage.rings.number_field.galois_group import GaloisGroup_v1
x = polygen(ZZ, 'x')
K = NumberField(x^3 + 2*x + 2, 'theta')
G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K)
H = G.group(); H
P = H.permutation_group(); P
sorted(P)

number_field()[source]¶

Return the number field of which this is the Galois group.

EXAMPLES:

Sage

sage: from sage.rings.number_field.galois_group import GaloisGroup_v1
sage: x = polygen(ZZ, 'x')
sage: K = NumberField(x^6 + 2, 't')
sage: G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K); G
...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
See https://github.com/sagemath/sage/issues/28782 for details.
Galois group PARI group [12, -1, 3, "D(6) = S(3)[x]2"] of degree 6 of the
 Number Field in t with defining polynomial x^6 + 2
sage: G.number_field()
Number Field in t with defining polynomial x^6 + 2

Python

>>> from sage.all import *
>>> from sage.rings.number_field.galois_group import GaloisGroup_v1
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(6) + Integer(2), 't')
>>> G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K); G
...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
See https://github.com/sagemath/sage/issues/28782 for details.
Galois group PARI group [12, -1, 3, "D(6) = S(3)[x]2"] of degree 6 of the
 Number Field in t with defining polynomial x^6 + 2
>>> G.number_field()
Number Field in t with defining polynomial x^6 + 2

Sage Live

from sage.rings.number_field.galois_group import GaloisGroup_v1
x = polygen(ZZ, 'x')
K = NumberField(x^6 + 2, 't')
G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K); G
G.number_field()

order()[source]¶

Return the order of this Galois group.

EXAMPLES:

Sage

sage: from sage.rings.number_field.galois_group import GaloisGroup_v1
sage: x = polygen(ZZ, 'x')
sage: K = NumberField(x^5 + 2, 'theta_1')
sage: G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K); G
...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
See https://github.com/sagemath/sage/issues/28782 for details.
Galois group PARI group [20, -1, 3, "F(5) = 5:4"] of degree 5 of the
 Number Field in theta_1 with defining polynomial x^5 + 2
sage: G.order()
20

Python

>>> from sage.all import *
>>> from sage.rings.number_field.galois_group import GaloisGroup_v1
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(5) + Integer(2), 'theta_1')
>>> G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K); G
...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
See https://github.com/sagemath/sage/issues/28782 for details.
Galois group PARI group [20, -1, 3, "F(5) = 5:4"] of degree 5 of the
 Number Field in theta_1 with defining polynomial x^5 + 2
>>> G.order()
20

Sage Live

from sage.rings.number_field.galois_group import GaloisGroup_v1
x = polygen(ZZ, 'x')
K = NumberField(x^5 + 2, 'theta_1')
G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K); G
G.order()

class sage.rings.number_field.galois_group.GaloisGroup_v2(number_field, algorithm='pari', names=None, gc_numbering=None, _type=None)[source]¶

Bases: GaloisGroup_perm

The Galois group of an (absolute) number field.

Note

We define the Galois group of a non-normal field \(K\) to be the Galois group of its Galois closure \(L\), and elements are stored as permutations of the roots of the defining polynomial of \(L\), not as permutations of the roots (in \(L\)) of the defining polynomial of \(K\). The latter would probably be preferable, but is harder to implement. Thus the permutation group that is returned is always simply-transitive.

The ‘arithmetical’ features (decomposition and ramification groups, Artin symbols etc) are only available for Galois fields.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group()
sage: G.subgroup([G([(1,2,3),(4,5,6)])])
Subgroup generated by [(1,2,3)(4,5,6)] of
 (Galois group 6T2 ([3]2) with order 6 of x^6 - 6*x^4 + 9*x^2 + 23)

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> G = NumberField(x**Integer(3) - x - Integer(1), 'a').galois_closure('b').galois_group()
>>> G.subgroup([G([(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5),Integer(6))])])
Subgroup generated by [(1,2,3)(4,5,6)] of
 (Galois group 6T2 ([3]2) with order 6 of x^6 - 6*x^4 + 9*x^2 + 23)

Sage Live

x = polygen(ZZ, 'x')
G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group()
G.subgroup([G([(1,2,3),(4,5,6)])])

Subgroups can be specified using generators (Issue #26816):

Sage

sage: K.<a> = NumberField(x^6 - 6*x^4 + 9*x^2 + 23)
sage: G = K.galois_group()
sage: list(G)
[(),
 (1,2,3)(4,5,6),
 (1,3,2)(4,6,5),
 (1,4)(2,6)(3,5),
 (1,5)(2,4)(3,6),
 (1,6)(2,5)(3,4)]
sage: g = G[1]
sage: h = G[3]
sage: list(G.subgroup([]))
[()]
sage: list(G.subgroup([g]))
[(), (1,2,3)(4,5,6), (1,3,2)(4,6,5)]
sage: list(G.subgroup([h]))
[(), (1,4)(2,6)(3,5)]
sage: sorted(G.subgroup([g,h])) == sorted(G)
True

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(6) - Integer(6)*x**Integer(4) + Integer(9)*x**Integer(2) + Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> G = K.galois_group()
>>> list(G)
[(),
 (1,2,3)(4,5,6),
 (1,3,2)(4,6,5),
 (1,4)(2,6)(3,5),
 (1,5)(2,4)(3,6),
 (1,6)(2,5)(3,4)]
>>> g = G[Integer(1)]
>>> h = G[Integer(3)]
>>> list(G.subgroup([]))
[()]
>>> list(G.subgroup([g]))
[(), (1,2,3)(4,5,6), (1,3,2)(4,6,5)]
>>> list(G.subgroup([h]))
[(), (1,4)(2,6)(3,5)]
>>> sorted(G.subgroup([g,h])) == sorted(G)
True

Sage Live

K.<a> = NumberField(x^6 - 6*x^4 + 9*x^2 + 23)
G = K.galois_group()
list(G)
g = G[1]
h = G[3]
list(G.subgroup([]))
list(G.subgroup([g]))
list(G.subgroup([h]))
sorted(G.subgroup([g,h])) == sorted(G)

Element[source]¶: alias of GaloisGroupElement

Subgroup[source]¶: alias of GaloisGroup_subgroup

artin_symbol(P)[source]¶

Return the Artin symbol \(\left(\frac{K / \QQ}{\mathfrak{P}}\right)\), where \(K\) is the number field of self, and \(\mathfrak{P}\) is an unramified prime ideal. This is the unique element \(s\) of the decomposition group of \(\mathfrak{P}\) such that \(s(x) = x^p \bmod \mathfrak{P}\), where \(p\) is the residue characteristic of \(\mathfrak{P}\).

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<b> = NumberField(x^4 - 2*x^2 + 2, 'a').galois_closure()
sage: G = K.galois_group()
sage: [G.artin_symbol(P) for P in K.primes_above(7)]
[(1,4)(2,3)(5,8)(6,7), (1,4)(2,3)(5,8)(6,7),
 (1,5)(2,6)(3,7)(4,8), (1,5)(2,6)(3,7)(4,8)]
sage: G.artin_symbol(17)
Traceback (most recent call last):
...
ValueError: Fractional ideal (17) is not prime
sage: QuadraticField(-7,'c').galois_group().artin_symbol(13)
(1,2)
sage: G.artin_symbol(K.primes_above(2)[0])
Traceback (most recent call last):
...
ValueError: Fractional ideal (...) is ramified

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(4) - Integer(2)*x**Integer(2) + Integer(2), 'a').galois_closure(names=('b',)); (b,) = K._first_ngens(1)
>>> G = K.galois_group()
>>> [G.artin_symbol(P) for P in K.primes_above(Integer(7))]
[(1,4)(2,3)(5,8)(6,7), (1,4)(2,3)(5,8)(6,7),
 (1,5)(2,6)(3,7)(4,8), (1,5)(2,6)(3,7)(4,8)]
>>> G.artin_symbol(Integer(17))
Traceback (most recent call last):
...
ValueError: Fractional ideal (17) is not prime
>>> QuadraticField(-Integer(7),'c').galois_group().artin_symbol(Integer(13))
(1,2)
>>> G.artin_symbol(K.primes_above(Integer(2))[Integer(0)])
Traceback (most recent call last):
...
ValueError: Fractional ideal (...) is ramified

Sage Live

x = polygen(ZZ, 'x')
K.<b> = NumberField(x^4 - 2*x^2 + 2, 'a').galois_closure()
G = K.galois_group()
[G.artin_symbol(P) for P in K.primes_above(7)]
G.artin_symbol(17)
QuadraticField(-7,'c').galois_group().artin_symbol(13)
G.artin_symbol(K.primes_above(2)[0])

complex_conjugation(P=None)[source]¶

Return the unique element of self corresponding to complex conjugation, for a specified embedding \(P\) into the complex numbers. If \(P\) is not specified, use the “standard” embedding, whenever that is well-defined.

EXAMPLES:

Sage

sage: L.<z> = CyclotomicField(7)
sage: G = L.galois_group()
sage: conj = G.complex_conjugation(); conj
(1,4)(2,5)(3,6)
sage: conj(z)
-z^5 - z^4 - z^3 - z^2 - z - 1

Python

>>> from sage.all import *
>>> L = CyclotomicField(Integer(7), names=('z',)); (z,) = L._first_ngens(1)
>>> G = L.galois_group()
>>> conj = G.complex_conjugation(); conj
(1,4)(2,5)(3,6)
>>> conj(z)
-z^5 - z^4 - z^3 - z^2 - z - 1

Sage Live

L.<z> = CyclotomicField(7)
G = L.galois_group()
conj = G.complex_conjugation(); conj
conj(z)

An example where the field is not CM, so complex conjugation really depends on the choice of embedding:

Sage

sage: x = polygen(ZZ, 'x')
sage: L = NumberField(x^6 + 40*x^3 + 1372, 'a')
sage: G = L.galois_group()
sage: [G.complex_conjugation(x) for x in L.places()]
[(1,3)(2,6)(4,5), (1,5)(2,4)(3,6), (1,2)(3,4)(5,6)]

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> L = NumberField(x**Integer(6) + Integer(40)*x**Integer(3) + Integer(1372), 'a')
>>> G = L.galois_group()
>>> [G.complex_conjugation(x) for x in L.places()]
[(1,3)(2,6)(4,5), (1,5)(2,4)(3,6), (1,2)(3,4)(5,6)]

Sage Live

x = polygen(ZZ, 'x')
L = NumberField(x^6 + 40*x^3 + 1372, 'a')
G = L.galois_group()
[G.complex_conjugation(x) for x in L.places()]

decomposition_group(P)[source]¶

Decomposition group of a prime ideal \(P\), i.e., the subgroup of elements that map \(P\) to itself. This is the same as the Galois group of the extension of local fields obtained by completing at \(P\).

This function will raise an error if \(P\) is not prime or the given number field is not Galois.

\(P\) can also be an infinite prime, i.e., an embedding into \(\RR\) or \(\CC\).

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^4 - 2*x^2 + 2, 'b').galois_closure()
sage: P = K.ideal([17, a^2])
sage: G = K.galois_group()
sage: G.decomposition_group(P)
Subgroup generated by [(1,8)(2,7)(3,6)(4,5)] of
 (Galois group 8T4 ([4]2) with order 8 of x^8 - 20*x^6 + 104*x^4 - 40*x^2 + 1156)
sage: G.decomposition_group(P^2)
Traceback (most recent call last):
...
ValueError: Fractional ideal (...) is not a prime ideal
sage: G.decomposition_group(17)
Traceback (most recent call last):
...
ValueError: Fractional ideal (17) is not a prime ideal

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(4) - Integer(2)*x**Integer(2) + Integer(2), 'b').galois_closure(names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal([Integer(17), a**Integer(2)])
>>> G = K.galois_group()
>>> G.decomposition_group(P)
Subgroup generated by [(1,8)(2,7)(3,6)(4,5)] of
 (Galois group 8T4 ([4]2) with order 8 of x^8 - 20*x^6 + 104*x^4 - 40*x^2 + 1156)
>>> G.decomposition_group(P**Integer(2))
Traceback (most recent call last):
...
ValueError: Fractional ideal (...) is not a prime ideal
>>> G.decomposition_group(Integer(17))
Traceback (most recent call last):
...
ValueError: Fractional ideal (17) is not a prime ideal

Sage Live

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^4 - 2*x^2 + 2, 'b').galois_closure()
P = K.ideal([17, a^2])
G = K.galois_group()
G.decomposition_group(P)
G.decomposition_group(P^2)
G.decomposition_group(17)

An example with an infinite place:

Sage

sage: x = polygen(ZZ, 'x')
sage: L.<b> = NumberField(x^3 - 2,'a').galois_closure(); G = L.galois_group()
sage: x = L.places()[0]
sage: G.decomposition_group(x).order()
2

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> L = NumberField(x**Integer(3) - Integer(2),'a').galois_closure(names=('b',)); (b,) = L._first_ngens(1); G = L.galois_group()
>>> x = L.places()[Integer(0)]
>>> G.decomposition_group(x).order()
2

Sage Live

x = polygen(ZZ, 'x')
L.<b> = NumberField(x^3 - 2,'a').galois_closure(); G = L.galois_group()
x = L.places()[0]
G.decomposition_group(x).order()

easy_order(algorithm=None)[source]¶

Return the order of this Galois group if it’s quick to compute.

EXAMPLES:

Sage

sage: R.<x> = ZZ[]
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 + 2*x + 2)
sage: G = K.galois_group()
sage: G.easy_order()
6
sage: x = polygen(ZZ, 'x')
sage: L.<b> = NumberField(x^72 + 2*x + 2)
sage: H = L.galois_group()
sage: H.easy_order()

Python

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(2)*x + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> G = K.galois_group()
>>> G.easy_order()
6
>>> x = polygen(ZZ, 'x')
>>> L = NumberField(x**Integer(72) + Integer(2)*x + Integer(2), names=('b',)); (b,) = L._first_ngens(1)
>>> H = L.galois_group()
>>> H.easy_order()

Sage Live

R.<x> = ZZ[]
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^3 + 2*x + 2)
G = K.galois_group()
G.easy_order()
x = polygen(ZZ, 'x')
L.<b> = NumberField(x^72 + 2*x + 2)
H = L.galois_group()
H.easy_order()

group()[source]¶

While GaloisGroup_v1 is being deprecated, this provides public access to the PARI/GAP group in order to keep all aspects of that API.

EXAMPLES:

Sage

sage: R.<x> = ZZ[]
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 + 2*x + 2)
sage: G = K.galois_group(type='pari')
...DeprecationWarning: the different Galois types have been merged into one class
See https://github.com/sagemath/sage/issues/28782 for details.
sage: G.group()
...DeprecationWarning: the group method is deprecated;
you can use _pol_galgp if you really need it
See https://github.com/sagemath/sage/issues/28782 for details.
PARI group [6, -1, 2, "S3"] of degree 3

Python

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(2)*x + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> G = K.galois_group(type='pari')
...DeprecationWarning: the different Galois types have been merged into one class
See https://github.com/sagemath/sage/issues/28782 for details.
>>> G.group()
...DeprecationWarning: the group method is deprecated;
you can use _pol_galgp if you really need it
See https://github.com/sagemath/sage/issues/28782 for details.
PARI group [6, -1, 2, "S3"] of degree 3

Sage Live

R.<x> = ZZ[]
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^3 + 2*x + 2)
G = K.galois_group(type='pari')
G.group()

inertia_group(P)[source]¶

Return the inertia group of the prime \(P\), i.e., the group of elements acting trivially modulo \(P\). This is just the 0th ramification group of \(P\).

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<b> = NumberField(x^2 - 3, 'a')
sage: G = K.galois_group()
sage: G.inertia_group(K.primes_above(2)[0])
Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 - 3)
sage: G.inertia_group(K.primes_above(5)[0])
Subgroup generated by [()] of (Galois group 2T1 (S2) with order 2 of x^2 - 3)

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(3), 'a', names=('b',)); (b,) = K._first_ngens(1)
>>> G = K.galois_group()
>>> G.inertia_group(K.primes_above(Integer(2))[Integer(0)])
Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 - 3)
>>> G.inertia_group(K.primes_above(Integer(5))[Integer(0)])
Subgroup generated by [()] of (Galois group 2T1 (S2) with order 2 of x^2 - 3)

Sage Live

x = polygen(ZZ, 'x')
K.<b> = NumberField(x^2 - 3, 'a')
G = K.galois_group()
G.inertia_group(K.primes_above(2)[0])
G.inertia_group(K.primes_above(5)[0])

is_galois()[source]¶

Whether the underlying number field is Galois.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: NumberField(x^3 - x + 1,'a').galois_group(names='b').is_galois()
False
sage: NumberField(x^2 - x + 1,'a').galois_group().is_galois()
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> NumberField(x**Integer(3) - x + Integer(1),'a').galois_group(names='b').is_galois()
False
>>> NumberField(x**Integer(2) - x + Integer(1),'a').galois_group().is_galois()
True

Sage Live

x = polygen(ZZ, 'x')
NumberField(x^3 - x + 1,'a').galois_group(names='b').is_galois()
NumberField(x^2 - x + 1,'a').galois_group().is_galois()

list()[source]¶

List of the elements of self.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: NumberField(x^3 - 3*x + 1,'a').galois_group().list()
[(), (1,2,3), (1,3,2)]

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> NumberField(x**Integer(3) - Integer(3)*x + Integer(1),'a').galois_group().list()
[(), (1,2,3), (1,3,2)]

Sage Live

x = polygen(ZZ, 'x')
NumberField(x^3 - 3*x + 1,'a').galois_group().list()

number_field()[source]¶

The ambient number field.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K = NumberField(x^3 - x + 1, 'a')
sage: K.galois_group(names='b').number_field() is K
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - x + Integer(1), 'a')
>>> K.galois_group(names='b').number_field() is K
True

Sage Live

x = polygen(ZZ, 'x')
K = NumberField(x^3 - x + 1, 'a')
K.galois_group(names='b').number_field() is K

order(algorithm=None, recompute=False)[source]¶

Return the order of this Galois group.

EXAMPLES:

Sage

sage: R.<x> = ZZ[]
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 + 2*x + 2)
sage: G = K.galois_group()
sage: G.order()
6

Python

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(2)*x + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> G = K.galois_group()
>>> G.order()
6

Sage Live

R.<x> = ZZ[]
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^3 + 2*x + 2)
G = K.galois_group()
G.order()

pari_label()[source]¶

Return the label assigned by PARI for this Galois group, an attempt at giving a human readable description of the group.

EXAMPLES:

Sage

sage: R.<x> = ZZ[]
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^8 - x^5 + x^4 - x^3 + 1)
sage: G = K.galois_group()
sage: G.transitive_label()
'8T44'
sage: G.pari_label()
'[2^4]S(4)'

Python

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(8) - x**Integer(5) + x**Integer(4) - x**Integer(3) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> G = K.galois_group()
>>> G.transitive_label()
'8T44'
>>> G.pari_label()
'[2^4]S(4)'

Sage Live

R.<x> = ZZ[]
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^8 - x^5 + x^4 - x^3 + 1)
G = K.galois_group()
G.transitive_label()
G.pari_label()

ramification_breaks(P)[source]¶

Return the set of ramification breaks of the prime ideal \(P\), i.e., the set of indices \(i\) such that the ramification group \(G_{i+1} \ne G_{i}\). This is only defined for Galois fields.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<b> = NumberField(x^8 - 20*x^6 + 104*x^4 - 40*x^2 + 1156)
sage: G = K.galois_group()
sage: P = K.primes_above(2)[0]
sage: G.ramification_breaks(P)
{1, 3, 5}
sage: min(G.ramification_group(P, i).order()
....:         / G.ramification_group(P, i + 1).order()
....:     for i in G.ramification_breaks(P))
2

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(8) - Integer(20)*x**Integer(6) + Integer(104)*x**Integer(4) - Integer(40)*x**Integer(2) + Integer(1156), names=('b',)); (b,) = K._first_ngens(1)
>>> G = K.galois_group()
>>> P = K.primes_above(Integer(2))[Integer(0)]
>>> G.ramification_breaks(P)
{1, 3, 5}
>>> min(G.ramification_group(P, i).order()
...         / G.ramification_group(P, i + Integer(1)).order()
...     for i in G.ramification_breaks(P))
2

Sage Live

x = polygen(ZZ, 'x')
K.<b> = NumberField(x^8 - 20*x^6 + 104*x^4 - 40*x^2 + 1156)
G = K.galois_group()
P = K.primes_above(2)[0]
G.ramification_breaks(P)
min(G.ramification_group(P, i).order()
        / G.ramification_group(P, i + 1).order()
    for i in G.ramification_breaks(P))

ramification_group(P, v)[source]¶

Return the \(v\)-th ramification group of self for the prime \(P\), i.e., the set of elements \(s\) of self such that \(s\) acts trivially modulo \(P^{(v+1)}\). This is only defined for Galois fields.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<b> = NumberField(x^3 - 3, 'a').galois_closure()
sage: G=K.galois_group()
sage: P = K.primes_above(3)[0]
sage: G.ramification_group(P, 3)
Subgroup generated by [(1,2,4)(3,5,6)] of
 (Galois group 6T2 ([3]2) with order 6 of x^6 + 243)
sage: G.ramification_group(P, 5)
Subgroup generated by [()] of (Galois group 6T2 ([3]2) with order 6 of x^6 + 243)

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(3), 'a').galois_closure(names=('b',)); (b,) = K._first_ngens(1)
>>> G=K.galois_group()
>>> P = K.primes_above(Integer(3))[Integer(0)]
>>> G.ramification_group(P, Integer(3))
Subgroup generated by [(1,2,4)(3,5,6)] of
 (Galois group 6T2 ([3]2) with order 6 of x^6 + 243)
>>> G.ramification_group(P, Integer(5))
Subgroup generated by [()] of (Galois group 6T2 ([3]2) with order 6 of x^6 + 243)

Sage Live

x = polygen(ZZ, 'x')
K.<b> = NumberField(x^3 - 3, 'a').galois_closure()
G=K.galois_group()
P = K.primes_above(3)[0]
G.ramification_group(P, 3)
G.ramification_group(P, 5)

signature()[source]¶

Return \(1\) if contained in the alternating group, \(-1\) otherwise.

EXAMPLES:

Sage

sage: R.<x> = ZZ[]
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: K.galois_group().signature()
-1
sage: K.<a> = NumberField(x^3 - 3*x - 1)
sage: K.galois_group().signature()
1

Python

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.galois_group().signature()
-1
>>> K = NumberField(x**Integer(3) - Integer(3)*x - Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.galois_group().signature()
1

Sage Live

R.<x> = ZZ[]
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^3 - 2)
K.galois_group().signature()
K.<a> = NumberField(x^3 - 3*x - 1)
K.galois_group().signature()

transitive_number(algorithm=None, recompute=False)[source]¶

Regardless of the value of gc_numbering, give the transitive number for the action on the roots of the defining polynomial of the original number field, not the Galois closure.

INPUT:

algorithm – string, specify the algorithm to be used
recompute – boolean, whether to recompute the result even if known by another algorithm

EXAMPLES:

Sage

sage: R.<x> = ZZ[]
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 + 2*x + 2)
sage: G = K.galois_group()
sage: G.transitive_number()
2
sage: x = polygen(ZZ, 'x')
sage: L.<b> = NumberField(x^13 + 2*x + 2)
sage: H = L.galois_group(algorithm='gap')
sage: H.transitive_number() # optional - gap_packages
9

Python

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(2)*x + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> G = K.galois_group()
>>> G.transitive_number()
2
>>> x = polygen(ZZ, 'x')
>>> L = NumberField(x**Integer(13) + Integer(2)*x + Integer(2), names=('b',)); (b,) = L._first_ngens(1)
>>> H = L.galois_group(algorithm='gap')
>>> H.transitive_number() # optional - gap_packages
9

Sage Live

R.<x> = ZZ[]
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^3 + 2*x + 2)
G = K.galois_group()
G.transitive_number()
x = polygen(ZZ, 'x')
L.<b> = NumberField(x^13 + 2*x + 2)
H = L.galois_group(algorithm='gap')
H.transitive_number() # optional - gap_packages

unrank(i)[source]¶

Return the \(i\)-th element of self.

INPUT:

i – integer between \(0\) and \(n-1\) where \(n\) is the cardinality of this set

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: G = NumberField(x^3 - 3*x + 1,'a').galois_group()
sage: [G.unrank(i) for i in range(G.cardinality())]
[(), (1,2,3), (1,3,2)]

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> G = NumberField(x**Integer(3) - Integer(3)*x + Integer(1),'a').galois_group()
>>> [G.unrank(i) for i in range(G.cardinality())]
[(), (1,2,3), (1,3,2)]

Sage Live

x = polygen(ZZ, 'x')
G = NumberField(x^3 - 3*x + 1,'a').galois_group()
[G.unrank(i) for i in range(G.cardinality())]