Opened 4 years ago

Closed 3 years ago

## #26816 closed enhancement (fixed)

# Specify subgroups of Galois groups using generators

Reported by: | pbruin | Owned by: | |
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Priority: | minor | Milestone: | sage-9.0 |

Component: | number fields | Keywords: | Galois group |

Cc: | Merged in: | ||

Authors: | Peter Bruin | Reviewers: | David Roe |

Report Upstream: | N/A | Work issues: | |

Branch: | 1c199f4 (Commits, GitHub, GitLab) | Commit: | 1c199f427f90c4013742fcef77bd3077e7893bf6 |

Dependencies: | Stopgaps: |

### Description (last modified by )

It should be possible to construct a subgroup of the Galois group of a number field from a set of generators, just as for permutation groups. This is already supported to some extent, but the list of elements of such a subgroup is wrong.

sage: R.<x> = QQ[] sage: f = x^6 - 6*x^4 + 9*x^2 + 23 sage: K.<a> = NumberField(f) 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]

Before:

sage: list(G.subgroup([])) [] sage: list(G.subgroup([g])) [(1,2,3)(4,5,6)] sage: list(G.subgroup([h])) [(1,4)(2,6)(3,5)] sage: list(G.subgroup([g,h])) [(1,2,3)(4,5,6), (1,4)(2,6)(3,5)]

After:

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: list(G.subgroup([g,h])) [(), (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)]

There is one user-visible change: the special case where applying `subgroup()`

to the list of all elements of `G`

returns `G`

itself has been removed. Besides simplifying the code, this has the advantage that the object returned by `subgroup()`

now always has type `GaloisGroup_subgroup`

, and in particular has a method `fixed_field()`

.

### Change History (5)

### comment:1 Changed 4 years ago by

Branch: | → u/pbruin/26816-GaloisGroup_subgroup |
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Commit: | → eed989369d086451145434d9e9b49841ce745332 |

Description: | modified (diff) |

Status: | new → needs_review |

### comment:2 Changed 4 years ago by

Commit: | eed989369d086451145434d9e9b49841ce745332 → 1c199f427f90c4013742fcef77bd3077e7893bf6 |
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### comment:3 Changed 3 years ago by

Reviewers: | → David Roe |
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Status: | needs_review → positive_review |

Looks good to me. Recent patchbot failures due to transition to Python 3.

### comment:4 Changed 3 years ago by

Milestone: | sage-8.5 → sage-9.0 |
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### comment:5 Changed 3 years ago by

Branch: | u/pbruin/26816-GaloisGroup_subgroup → 1c199f427f90c4013742fcef77bd3077e7893bf6 |
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Resolution: | → fixed |

Status: | positive_review → closed |

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Branch pushed to git repo; I updated commit sha1. New commits:

`Merge branch 'develop' into ticket/26816-GaloisGroup_subgroup`