On Wednesday, September 4, 2024 at 1:41:30 PM UTC+9 Nils Bruin wrote:

.. Indeed, Magma's algebraic closure of QQ does not use an embedding into 
CC but instead tracks conjugate labels using a large finite field.


You mean a large number field? 

>From QQbar doc:

As with many other implementations of the algebraic numbers, we try
hard to avoid computing a number field and working in the number
field; instead, we use floating-point interval arithmetic whenever
possible (basically whenever we need to prove non-equalities), and
resort to symbolic computation only as needed (basically to prove
equalities).   

Thus QQbar assumes a complex embedding from the ground, perhaps unlike 
Magma.

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