complex_embedding on relative number fields is inconsistent with the base field

[edgarcosta]

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

Component: number fields

Keywords: complex_embedding relative

Issue created by migration from https://trac.sagemath.org/ticket/22008

[sagetrac-bober]

comment:1

sage: L.absolute_field('z')
Number Field in z with defining polynomial x^2 - x - 1
sage: M.absolute_field('z')
Number Field in z with defining polynomial x^2 + x - 1

[videlec]

comment:3

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.