Implement the universal cyclotomic field, using Zumbroich basis

[nthiery]

This patch provides the universal cyclotomic field

    sage: UCF.<E> = UniversalCyclotomicField(); 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.

Depends on #13765

CC: @sagetrac-sage-combinat @sagetrac-cwitty

Component: number fields

Keywords: Cyclotomic field, Zumbroich basis

Author: Christian Stump, Simon King

Reviewer: Frédéric Chapoton

Merged: sage-5.7.beta3

Issue created by migration from https://trac.sagemath.org/ticket/8327

[craigcitro]

comment:2

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)?

[nthiery]

comment:3

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.

[craigcitro]

comment:4

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.

[nthiery]

comment:5

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?

[stumpc5]

comment:6

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.

[stumpc5]

comment:7

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)?

• how can I embed a CombinatorialAlgebra into another one (here: NewCyclotomicField(m) into NewCyclotomicField(n) for m | n)?

• how can I define a new class UniversalCyclotomicField containing all cyclotomic fields?

• how can I define the attribute self._one in the constructor of NewCyclotomicField properly?

[stumpc5]

Author: Christian Stump

[stumpc5]

Description changed:

--- 
+++ 
@@ -1,58 +1,56 @@
-Here is a user story for this feature.
-
-We construct the universal cyclotomic field::
+This patch provides the universal cyclotomic field
 
 ```
-    sage: F = CyclotomicField()
+    sage: UCF
+    Universal Cyclotomic Field endowed with the Zumbroich basis
 ```
 
-This field contains all roots of unity:
+in sage. This field is the smallest field extension of QQ which contains all roots of unity.
 
 ```
-    sage: z3 = F.zeta(3)
-    sage: z3
+    sage: E(3); E(3)^3
     E(3)
-    sage: z3^3
     1
-    sage: z5 = F.zeta(5)
-    sage: z5
-    E(5)
-    sage: z5^5
+    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 consists of a strict subset of all roots of unity::
+basis, which gives, for any n, A basis of QQ( E(n) ) over QQ, where (n,k) stands for E(n)^k.
 
 ```
-    sage: z9 = F.zeta(9)
-    -E(9)^4-E(9)^7
-    sage: z3 * z5
-    sage: E(15)^8
-    sage: z3 + z5
-    -E(15)^2-2*E(15)^8-E(15)^11-E(15)^13-E(15)^14
-    sage: [z9^i for i in range(0,9)]
-    [1, -E(9)^4-E(9)^7, E(9)^2, E(3), E(9)^4, E(9)^5, E(3)^2, E(9)^7, -E(9)^2-E(9)^5 ]
+    sage: UCF.zumbroich_basis(6)
+    [(6, 2), (6, 4)]
 ```
 
-Note: we might want some other style of pretty printing.
-
-The following is called AsRootOfUnity in Chevie; we might want instead
-to use (z1*z3).multiplicative_order()::
+As seen for E(6), every element in UCF is expressed in terms of the smallest cyclotomic field in which it is contained.
 
 ```
-    sage: (z1*z3).as_root_of_unity()
-    11/18
+sage: E(6)*E(4)
+-E(12)^11
 ```
 
-Depending on the progress on #6391 (lib gap), we might want to
-implement this directly in Sage or to instead expose GAP's
-implementation, creating elements as in::
+It provides arithmetics on UCF as addition, multiplication, and inverses:
 
 ```
-sage: z5 = gap("E(5)")
-sage: z3 = gap("E(3)")
-sage: z3+z5
--E(15)^2-2*E(15)^8-E(15)^11-E(15)^13-E(15)^14
+    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).