#8327: Implement the universal cyclotomic field, using Zumbroich basis
-----------------------------+----------------------------------------------
Reporter: nthiery | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.0
Component: number fields | Keywords: Cyclotomic field, Zumbroich
basis
Work_issues: | Upstream: N/A
Reviewer: | Author: Christian Stump, Simon King
Merged: | Dependencies: #4539 #10771
-----------------------------+----------------------------------------------
Changes (by davidloeffler):
* dependencies: #10771 => #4539 #10771
Old description:
> 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__
>
> * [attachment:trac_8327_universal_cyclotomic_field-cs.patch]
New description:
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__
* [attachment:trac_8327-rebased_for_v5.patch]
--
Comment:
Apply trac_8327-rebased_for_v5.patch
(for the patchbot).
The previous patch conflicts with #4539, so I have done a new rebased
patch. There was one change in {{{free_module.py}}} which was rejected,
and whose purpose was not at all clear to me, so I didn't include it in
the rebased patch -- all doctests seem to pass without it.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8327#comment:71>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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