Galois groups of field extensions.

We do not necessarily require extensions to be normal, but we do require them to be separable.

When an extension is not normal, the Galois group refers to the automorphism group of the normal closure.

AUTHORS:

• David Roe (2019): initial version

class sage.groups.galois_group._GaloisMixin

This class provides methods for Galois groups, allowing concrete instances to inherit from both permutation group and abelian group classes.

class sage.groups.galois_group._SubGaloisMixin

This class provides methods for subgroups of Galois groups, allowing concrete instances to inherit from both permutation group and abelian group classes.

class sage.groups.galois_group.GaloisGroup_ab(field, generator_orders, algorithm=None, gen_names='sigma')

Bases: _GaloisMixin, AbelianGroup_class

Abelian Galois groups

Subgroup

alias of GaloisSubgroup_ab

is_galois()

Abelian extensions are Galois.

For compatibility with Galois groups of number fields.

EXAMPLES:

Sage

sage: GF(9).galois_group().is_galois()                                      # needs sage.rings.finite_rings
True

Python

>>> from sage.all import *
>>> GF(Integer(9)).galois_group().is_galois()                                      # needs sage.rings.finite_rings
True

Sage Live

GF(9).galois_group().is_galois()                                      # needs sage.rings.finite_rings

permutation_group()

Return a permutation group giving the action on the roots of a defining polynomial.

This is the regular representation for the abelian group, which is not necessarily the smallest degree permutation representation.

EXAMPLES:

Sage

sage: GF(3^10).galois_group().permutation_group()                           # needs sage.libs.gap sage.rings.finite_rings
Permutation Group with generators [(1,2,3,4,5,6,7,8,9,10)]

Python

>>> from sage.all import *
>>> GF(Integer(3)**Integer(10)).galois_group().permutation_group()                           # needs sage.libs.gap sage.rings.finite_rings
Permutation Group with generators [(1,2,3,4,5,6,7,8,9,10)]

Sage Live

GF(3^10).galois_group().permutation_group()                           # needs sage.libs.gap sage.rings.finite_rings

transitive_number(algorithm=None, recompute=False)

Return the transitive number for the action on the roots of the defining polynomial.

For abelian groups, there is only one transitive action up to isomorphism (left multiplication of the group on itself), so we identify that action.

EXAMPLES:

Sage

sage: from sage.groups.galois_group import GaloisGroup_ab
sage: Gtest = GaloisGroup_ab(field=None, generator_orders=(2,2,4))
sage: Gtest.transitive_number()                                             # needs sage.libs.gap
2

Python

>>> from sage.all import *
>>> from sage.groups.galois_group import GaloisGroup_ab
>>> Gtest = GaloisGroup_ab(field=None, generator_orders=(Integer(2),Integer(2),Integer(4)))
>>> Gtest.transitive_number()                                             # needs sage.libs.gap
2

Sage Live

from sage.groups.galois_group import GaloisGroup_ab
Gtest = GaloisGroup_ab(field=None, generator_orders=(2,2,4))
Gtest.transitive_number()                                             # needs sage.libs.gap

class sage.groups.galois_group.GaloisGroup_cyc(field, generator_orders, algorithm=None, gen_names='sigma')

Bases: GaloisGroup_ab

Cyclic Galois groups

signature()

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

EXAMPLES:

Sage

sage: GF(3^2).galois_group().signature()                                    # needs sage.rings.finite_rings
-1
sage: GF(3^3).galois_group().signature()                                    # needs sage.rings.finite_rings
1

Python

>>> from sage.all import *
>>> GF(Integer(3)**Integer(2)).galois_group().signature()                                    # needs sage.rings.finite_rings
-1
>>> GF(Integer(3)**Integer(3)).galois_group().signature()                                    # needs sage.rings.finite_rings
1

Sage Live

GF(3^2).galois_group().signature()                                    # needs sage.rings.finite_rings
GF(3^3).galois_group().signature()                                    # needs sage.rings.finite_rings

transitive_number(algorithm=None, recompute=False)

Return the transitive number for the action on the roots of the defining polynomial.

EXAMPLES:

Sage

sage: GF(2^8).galois_group().transitive_number()                            # needs sage.rings.finite_rings
1
sage: GF(3^32).galois_group().transitive_number()                           # needs sage.rings.finite_rings
33
sage: GF(2^60).galois_group().transitive_number()                           # needs sage.rings.finite_rings
Traceback (most recent call last):
...
NotImplementedError: transitive database only computed up to degree 47

Python

>>> from sage.all import *
>>> GF(Integer(2)**Integer(8)).galois_group().transitive_number()                            # needs sage.rings.finite_rings
1
>>> GF(Integer(3)**Integer(32)).galois_group().transitive_number()                           # needs sage.rings.finite_rings
33
>>> GF(Integer(2)**Integer(60)).galois_group().transitive_number()                           # needs sage.rings.finite_rings
Traceback (most recent call last):
...
NotImplementedError: transitive database only computed up to degree 47

Sage Live

GF(2^8).galois_group().transitive_number()                            # needs sage.rings.finite_rings
GF(3^32).galois_group().transitive_number()                           # needs sage.rings.finite_rings
GF(2^60).galois_group().transitive_number()                           # needs sage.rings.finite_rings

class sage.groups.galois_group.GaloisSubgroup_ab(ambient, gens, names='f', category=None)

Bases: AbelianGroup_subgroup, _SubGaloisMixin

Subgroups of abelian Galois groups.

Make live