padic folder refactoring

[roed314]

The p-adic directory has a lot of files, and there's a lot of boilerplate code necessary to write new parents. This is a (meta)-ticket to track a number of improvements to improve the situation.

• Use dynamic classes to create parents. This would be done in factory.py and would mix in a class for the precision type, a class for the ramification type (base, unramified, eisenstein or general), a class for ring vs field, etc. See Use dynamic classes for p-adic parents #22054

• There are some files (e.g. qadic_flint_CR.pyx) which exist just because Cython needs a file to compile. The actual code is all in templates and linkage files. These should be moved to a subdirectory. This would also be a good time to do any other renaming of files (see Change file names in sage/rings/padics #14324)

• The factory file should be simplified. There is a lot of duplicated documentation, and the processing of the arguments to the factories is hard to understand.

This should wait on various tickets to be merged, in order to alleviate merge conflicts:

• cannot coerce p-adic field into residue field #8240
• Factoring padic polynomials #12561
• Better implementation of Frobenius automorphism #12657
• Improve padic add_bigoh #13591
• Iterators for p-adic expansions, polynomial representations of padic elements #14825
• p-adic extensions with non-monic polynomials #20073
• root_field() does not work for p-adic fields #20244
• p-adic polylogarithms, after Besser-de Jeu #20260
• change for p-adic rings #20310
• Printing p-adic numbers #22103
• p-adic completions/Henselizations backed by exact rings #22956 (maybe)
• Ramified extensions of general p-adic rings and fields #23218
• Fix modulo operator for unramified floating point elements #23228
• Cache fraction_field() of p-adic rings, deprecate print_mode options #23229
• Fast p-adic exponential #23235
• Allow exact defining polynomials for p-adic extensions #23331
• Fast logarithm for p-adic floats #23449
• Smith form of p-adic matrices #23450
• Conversion between p-adic rings and fields with the same defining polynomial #23471
• Determinant of p-adic matrices #23478
• Echelon form for p-adic matrices #23486
• Lattice precision for p-adics #23505 (maybe)
• Exact ring attached to a p-adic ring or field #23507
• Improvements for p-adic polynomials #23508
• Fraction field of fixed modulus p-adic rings should have floating point type #23510

The following tickets should be okay because they only change files unaffected by these proposals:

• Artin-Hasse exponential #12560
• Convergent series on ultrametric discs #14830
• Toy implementation of Matrix-F5 algorithm and various adaptation #20997
• Compute Gauss sums with Gross-Koblitz formula #23456
• Method denominator() for elements in CDVF #23503

The following tickets are specific to NTL implementations, and should maybe be abandoned

• padic _ntl_rep_abs incorrect for extension fields #13651
• matrix_mod_pn incorrect over extensions of padics #13659
• accuracy problem in matrix_mod_pn for Eisenstein extensions of padics #13661

The following tickets have branches that will be in merge hell, but we propose to abandon.

• Representations and latex representations of uniformizers #14829

CC: @saraedum

Component: padics

Keywords: sd87

Issue created by migration from https://trac.sagemath.org/ticket/23524