#8327: Implement the universal cyclotomic field, using Zumbroich basis
------------------------------------------------+---------------------------
Reporter: nthiery | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.8
Component: number fields | Keywords: Cyclotomic
field, Zumbroich basis
Work_issues: Rebase wrt #10771. Fix doctests. | Upstream: N/A
Reviewer: | Author: Christian
Stump, Simon King
Merged: | Dependencies: #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_universal_cyclotomic_field-cs.patch]
* [attachment:trac8327_lazy_import_UCF.patch]
--
Comment(by SimonKing):
OK, done.
When applying both patches, one gets
{{{
sage: 'numpy' in sys.modules
False
sage: sage0("'numpy' in sys.modules")
False
sage: UCF
Universal Cyclotomic Field endowed with the Zumbroich basis
sage: 'numpy' in sys.modules
True
sage: sage0("'numpy' in sys.modules")
False
}}}
So, numpy is only imported when needed, and this is what #11714 was about.
Apply trac_8327_universal_cyclotomic_field-cs.patch
trac8327_lazy_import_UCF.patch
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8327#comment:53>
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