Comparing numbers in an algebraic field

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What bothers me is that, though Qb is defined with an embedding in $\mathbb{C}$, there is no canonical coercion from Qb to QQbar, as if the information of the embedding was lost (ideally, there should be a difference between "number field" and "embedded number field", even two different parents):

sage: cm = sage.structure.element.get_coercion_model()
sage: cm.common_parent(Qb,QQbar)
TypeError: no common canonical parent for objects with parents: 'Number Field in b with defining polynomial x^3 - 2*x^2 - 2*x - 2' and 'Algebraic Field'

Not even a conversion:

sage: QQbar(b)
TypeError: Illegal initializer for algebraic number
sage: AA(b)
TypeError: Illegal initializer for algebraic number

That said, the following workaround to consider b as a real algebraic number could do the job:

sage: Qb.embeddings(AA)
[
Ring morphism:
  From: Number Field in b with defining polynomial x^3 - 2*x^2 - 2*x - 2
  To:   Algebraic Real Field
  Defn: b |--> 2.919639565839419?
]
sage: bb = Qb.embeddings(AA)[0](b) ; bb
2.919639565839419?
sage: 0 < bb
True

Comparison in number fields is not satisfactory in Sage.

The current implementation (except in some special cases) seems to be that

• > always returns True,
• < always returns False.

That is the case with b defined as in your example:

sage: b > 0
True
sage: 0 > b
True
sage: b < 0
False
sage: 0 < b
False

Comparison works well in quadratic number fields since ticket #13213.

The method is_real_positive() from ticket #8347 works in some number fields (but unfortunately not in the field Qb in your question).