Opened 10 years ago

Last modified 7 years ago

#8327 closed enhancement

Implement the universal cyclotomic field, using Zumbroich basis — at Version 28

Reported by: nthiery Owned by:
Priority: major Milestone: sage-5.7
Component: number fields Keywords: Cyclotomic field, Zumbroich basis
Cc: sage-combinat, cwitty Merged in:
Authors: Christian Stump, Simon King Reviewers:
Report Upstream: N/A Work issues: needs more doctests and Sphinxified docstrings
Branch: Commit:
Dependencies: Stopgaps:

Description (last modified by SimonKing)

This patch provides the universal cyclotomic field

    sage: UCF
    Universal Cyclotomic Field endowed with the Zumbroich basis

in sage. This field is the smallest field extension of QQ which contains all roots of unity.

    sage: E(3); E(3)^3
    E(3)
    1
    sage: E(6); E(6)^2; E(6)^3; E(6)^6
    -E(3)^2
    E(3)
    -1
    1

It comes equipped with a distinguished basis, called the Zumbroich basis, which gives, for any n, A basis of QQ( E(n) ) over QQ, where (n,k) stands for E(n)k.

    sage: UCF.zumbroich_basis(6)
    [(6, 2), (6, 4)]

As seen for E(6), every element in UCF is expressed in terms of the smallest cyclotomic field in which it is contained.

sage: E(6)*E(4)
-E(12)^11

It provides arithmetics on UCF as addition, multiplication, and inverses:

    sage: E(3)+E(4)
    E(12)^4 - E(12)^7 - E(12)^11
    sage: E(3)*E(4)
    E(12)^7
    sage: (E(3)+E(4)).inverse()
    E(12)^4 + E(12)^8 + E(12)^11
    sage: (E(3)+E(4))*(E(3)+E(4)).inverse()
    1

And also things like Galois conjugates.

    sage: (E(3)+E(4)).galois_conjugates()
    [E(12)^4 - E(12)^7 - E(12)^11, -E(12)^7 + E(12)^8 - E(12)^11, E(12)^4 + E(12)^7 + E(12)^11, E(12)^7 + E(12)^8 + E(12)^11]

The ticket does not use the gap interface; it depends on #9651 (Addition of combinatorial free module).

Apply

Change History (28)

comment:1 Changed 10 years ago by nthiery

  • Cc sage-combinat added

comment:2 follow-up: Changed 10 years ago by craigcitro

So this seems interesting (I'd never heard of the Zumbrioch basis before). I don't have anything useful to say about the utility of this, or implementing it.

However, I'd like to suggest we might want to use a different convention for the constructor -- I don't like the idea of CyclotomicField() working. That seems like something that's too easy for a user to accidentally do (leave out the n they intended to use), and I'd much rather they see an error than have it quietly succeed, only to discover that their cyclotomic field isn't quite what they expected (and likely slower to boot). Maybe something like F = CyclotomicField(n=Infinity)?

comment:3 in reply to: ↑ 2 ; follow-up: Changed 10 years ago by nthiery

Replying to craigcitro:

So this seems interesting (I'd never heard of the Zumbrioch basis before). I don't have anything useful to say about the utility of this, or implementing it.

However, I'd like to suggest we might want to use a different convention for the constructor -- I don't like the idea of CyclotomicField() working. That seems like something that's too easy for a user to accidentally do (leave out the n they intended to use), and I'd much rather they see an error than have it quietly succeed, only to discover that their cyclotomic field isn't quite what they expected (and likely slower to boot). Maybe something like F = CyclotomicField(n=Infinity)?

I indeed take any better suggestion! In Gap that would be Cyclotomics, but it does sound good in Field in the name. n=infinity might be misleading, since it's not about infinite roots of unity.

comment:4 in reply to: ↑ 3 ; follow-up: Changed 10 years ago by craigcitro

Replying to nthiery:

I indeed take any better suggestion! In Gap that would be Cyclotomics, but it does sound good in Field in the name. n=infinity might be misleading, since it's not about infinite roots of unity.

True. Maybe it's a bit too "cutesy," but using n=0 might be nice -- after all, the nth cyclotomic field has roots of unity for all divisors of n, so this would still hold for the universal cyclotomic field and n=0.

comment:5 in reply to: ↑ 4 Changed 10 years ago by nthiery

Replying to craigcitro:

True. Maybe it's a bit too "cutesy," but using n=0 might be nice -- after all, the nth cyclotomic field has roots of unity for all divisors of n, so this would still hold for the universal cyclotomic field and n=0.

Mmm, any complex number is a 0-th root of unity, isn't it?

comment:6 Changed 10 years ago by stumpc5

  • Status changed from new to needs_work

Does anyone have a pdf copy of Zumbroich's thesis where the algorithms are described?

There is a nice version of a modified Zumbroich basis in the article I used (both are implemented in the attached file) but I don't see how the author expresses any root of unity in terms of the basis (probably just due to my lack of knowledge). But Zumbroich should describe it in more detail in his thesis.

comment:7 Changed 10 years ago by stumpc5

The attached class NewCyclotomicField? uses the modified Zumbroich basis and behaves already somehow as it should (beside pretty printing).

If people think the Zumbroich basis itself is nicer, it's easy to change. I personally think the modified version as defined in Breuer - "Integral bases for subfields of cyclotomic fields" AAECC8, 279--289 (1997) has nicer properties (see the definition of the set S_n and Remark 1).

Here are some implementation problems I have:

  • how can I replace CombinatorialAlgebra? by a new field constructor (I haven't found a description how to define a field)?

comment:8 Changed 10 years ago by cwitty

  • Cc cwitty added

comment:9 Changed 10 years ago by stumpc5

  • Status changed from needs_work to needs_review

comment:10 Changed 10 years ago by stumpc5

  • Authors set to Christian Stump

comment:11 Changed 9 years ago by stumpc5

  • Description modified (diff)

comment:12 Changed 9 years ago by davidloeffler

  • Milestone set to sage-feature
  • Status changed from needs_review to needs_work

Apply trac_8327_universal_cyclotomic_field-cs.2.patch

(for patchbot -- I hope that's right!).

This doesn't quite look ready to me. I haven't tried applying the patch, but just on a visual check, I can see that lots of methods aren't documented or have no tests (*every* function added to Sage must be doctested, even those whose entire body is "raise NotImplementedError"). Also, the new module should be added to the reference manual, and the formatting should be valid ReST markup.

I very much hope you can get this code into shape -- it'd be a great thing to have in Sage -- but for now I'm afraid it's a thumbs down.

comment:13 follow-up: Changed 9 years ago by davidloeffler

  • Work issues set to Won't build, needs more doctests and Sphinxified docstrings

I've looked at the patchbot logs, and it gets worse -- the Cython code isn't valid apparently.

comment:14 in reply to: ↑ 13 Changed 9 years ago by stumpc5

Replying to davidloeffler:

I've looked at the patchbot logs, and it gets worse -- the Cython code isn't valid apparently.

This was only as cython booleans changed in the meantime. Beside that, the code was always working properly. I changed this minor problem, and it should be working now again.

I improved the documentation but there are still some things to be done...

comment:15 Changed 9 years ago by stumpc5

  • Status changed from needs_work to needs_review

For the buildbot:

Apply trac_8327_universal_cyclotomic_field-cs.patch

All other files are obsolete, but I cannot delete them. Some documentation is still missing, but only for methods which are helper functions for others.

Would be nice to get some feedback!

comment:16 Changed 9 years ago by stumpc5

Can some delete all but the youngest attached patch, the buildbot gets confused.

Thanks, Christian

comment:17 Changed 9 years ago by stumpc5

  • Description modified (diff)

comment:18 Changed 9 years ago by davidloeffler

Apply trac_8327_universal_cyclotomic_field-cs.patch

comment:19 Changed 9 years ago by stumpc5

Apply only trac_8327_universal_cyclotomic_field-cs.patch

comment:20 Changed 9 years ago by stumpc5

  • Work issues changed from Won't build, needs more doctests and Sphinxified docstrings to needs more doctests and Sphinxified docstrings

Apply only trac_8327_universal_cyclotomic_field-cs.patch

comment:21 Changed 9 years ago by stumpc5

Apply only trac_8327_universal_cyclotomic_field-cs.patch

comment:22 Changed 9 years ago by chapoton

Apply trac_8327_universal_cyclotomic_field-cs.patch

comment:23 Changed 9 years ago by stumpc5

  • Dependencies set to #10963
  • Owner changed from davidloeffler to (none)

comment:24 Changed 9 years ago by stumpc5

  • Milestone changed from sage-feature to sage-4.7.2

comment:25 Changed 8 years ago by stumpc5

  • Dependencies #10963 deleted

comment:26 follow-up: Changed 8 years ago by nthiery

  • Status changed from needs_review to needs_work

When applying the sage-combinat queue on sage 4.8.alpha0, we got a compilation error on universal_cyclotomic_field_c.pyx, most likely due to the updated cython (#11761)

comment:27 in reply to: ↑ 26 Changed 8 years ago by SimonKing

  • Authors changed from Christian Stump to Christian Stump, Simon King
  • Status changed from needs_work to needs_review

Replying to nthiery:

When applying the sage-combinat queue on sage 4.8.alpha0, we got a compilation error on universal_cyclotomic_field_c.pyx, most likely due to the updated cython (#11761)

My patch seems to fix the problem. At least, when one uses the new Cython spkg and applies the combinat queue up to and including Christian's patch and adds my patch as well (which is also in the combinat queue, I think), then sage builds and starts.

Note that the my patch does not add the new Cython as a dependency - it should run with the old Cython as well.

However, I already get a doctest error in devel/sage-combinat/doc/en/thematic_tutorials/sandpile.rst. Now it is needed to test whether this error is due to the patches from here, or due to some other patch in the combinat queue.

Since it is not sure whether THIS ticket is the culprit, and since it builds with the new cython, I am promiting it to "needs review".

Apply trac_8327_universal_cyclotomic_field-cs.patch trac_8327_universal_cyclotomic_field_new_cython-sk.patch

comment:28 Changed 8 years ago by SimonKing

  • Description modified (diff)

Sorry, I forgot to change the ticket description.

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