Cleanup and speedup in QQbar

[vdelecroix]

This is a task ticket for speed up and cleaning in sage.rings.qqbar

Prerequisites on number fields/polynomials/interval arithmetic

• #18332: implement is_integer and is_rational on number field elements
• #18334: implement unique_sign and unique_trunc on interval field elements
• #18337: implement real and imag on real intervals
• #17830: consider the natural order induced from RR for number fields with real embedding
• #19362: refine root
• #18356: implement composed_op from [BFSS]
• #19824: faster comparison for (real) embedded number fields
• #24503: enhanced LazyAlgebraic

Tasks in AA/QQbar:

• #18303: better comparisons
• #17886, #18242: faster exactification operations using power series and resultants (see also #18356)
• #15600: exactification is slow in do_polred
• #16222, #18122: enhanced minpoly
• #19954 and #19955: remove all descriptors except ANRoot, ANExtension and ANRational
• #20074: simplify method dispatcher for binary operators
• #12745: conversion issue QQbar -> AA
• #18106: introduce a polynomial descriptor, make sum and product be n-ary instead of binary and get rid of recursive calls
• #12715: number field embeddings should always have target in AA or QQbar
• #19356: QQbar.polynomial_root(): allow approximate root
• #21095: slowness in exactification
• #21838: do not make appear the imaginary (or real) part in QQbar when

it is known to be zero

• #5574: Implement QQbar^{QQ as action}
• use firstly interval arithmetic for sign/floor/ceil/trunc/round instead of the _floor_ceil helper
• remove the following methods from ANDescr:
  • rational_value
  • is_field_element
  • is_exact
• (?) enhanced sage_input for ANExtension
• better powering (__pow__) using ANExtension and fix convention for powering in AA vs QQbar
• reimplement _real_refine_interval without these ugly dictionaries (see also #17895)
• use directly embedded number fields in AlgebraicGenerator as comparison there is faster (see #17830 and #19824). We might want that these embedded number field carries an interval of fixed precision with them (let say 64) to avoid reevaluation in most cases.

Bugs

• #20209: QQbar -> RIF conversion fails
• #12745: Coercion problem with Algebraic Real Field
• #20288: representation and mutability of the description

[zimmerma]

about slowness of QQbar, see #19910.

[nbruin]

There's an issue on QQBar element powering reported in ​#18836 comment 9 that may have interaction with what you're doing here (perhaps you're solving the issue?). Perhaps it's worthwhile including that problem in this QQbar cleanup operation somewhere.