experimentally: allow small hight lifts

[fieker]

julia> k, a = wildanger_field(3, 13) (Number field of degree 3 over QQ, _$)

julia> c, mc = Hecke.unramified_completion(k, prime_ideals_over(maximal_order(k), 17)[1]) (Unramified extension of 17-adic numbers of degree 1, Map: k -> c)

julia> preimage(mc, mc(a)) 2639044339939425523644123345123996041787685738888204449072451948418018872324117

julia> preimage(mc, mc(a); small_lift = true, integral = true) _$

julia> preimage(mc, mc(a^3//5-2); small_lift = true) 13//5*$^2 - 13//5*$ + 3//5

(and the same for plain completion)

needs tests and polish, but Simon wanted to play

[codecov[bot]]

Codecov Report

Attention: Patch coverage is 97.14286% with 1 lines in your changes are missing coverage. Please review.

Project coverage is 75.36%. Comparing base (f9a2779) to head (86e6b6b). Report is 3 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1506      +/-   ##
==========================================
- Coverage   75.37%   75.36%   -0.02%     
==========================================
  Files         354      354              
  Lines      112723   112811      +88     
==========================================
+ Hits        84966    85021      +55     
- Misses      27757    27790      +33     

Files	Coverage Δ
src/LocalField/Types.jl	100.00% <ø> (ø)
src/LocalField/Completions.jl	96.56% <97.14%> (+0.27%)	:arrow_up:

... and 33 files with indirect coverage changes

[simonbrandhorst]

Thank you! (Edit: The example works now, it seems that revise did not update the type definition)

[thofma]

Can someone supply an example? Then we can add a test and get this in.

[simonbrandhorst]

This seems to work for unramified extensions, but for ramified ones not. And there is another bug for setprecision! documented as a broken test @joschmitt