Opened 6 years ago

Last modified 7 days ago

## #22008 new defect

# complex_embedding on relative number fields is inconsistent with the base field

Reported by: | Edgar Costa | Owned by: | |
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Priority: | major | Milestone: | sage-9.8 |

Component: | number fields | Keywords: | complex_embedding relative |

Cc: | Merged in: | ||

Authors: | Reviewers: | ||

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

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### Description (last modified by )

Here is a minimal example, with a trivial extension.

QQx.<x> = QQ[] L.<b> = NumberField(x^2 - x - 1) Ly.<y> = L[]; M.<c> = NumberField(y) print(L(b).complex_embedding()) print(M(b).complex_embedding()) -0.618033988749895 1.61803398874989

### Change History (7)

### comment:2 Changed 17 months ago by

Description: | modified (diff) |
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Milestone: | sage-7.5 → sage-9.4 |

### comment:3 Changed 17 months ago by

As written in the documentation, the function `complex_embedding`

returns the image of the `i`

-th embedding into the complex numbers (default is `i=0`

). Here is the list

sage: L.complex_embeddings() [ Ring morphism: From: Number Field in b with defining polynomial x^2 - x - 1 with b = 1.618033988749895? To: Complex Field with 53 bits of precision Defn: b |--> -0.618033988749895, Ring morphism: From: Number Field in b with defining polynomial x^2 - x - 1 with b = 1.618033988749895? To: Complex Field with 53 bits of precision Defn: b |--> 1.61803398874989 ]

In particular

`i=0`

is not the complex embedding coming from the coercion embedding- for extension fields this list is not sorted according to the embedding of the base

I definitely agree that this is confusing. To my mind, the function `b.complex_embedding()`

is to blame and it is a mistake to have made `i=0`

the default.

### comment:4 Changed 13 months ago by

Milestone: | sage-9.4 → sage-9.5 |
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### comment:5 Changed 9 months ago by

Milestone: | sage-9.5 → sage-9.6 |
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### comment:6 Changed 5 months ago by

Milestone: | sage-9.6 → sage-9.7 |
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### comment:7 Changed 7 days ago by

Milestone: | sage-9.7 → sage-9.8 |
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