Fix gauss_sum() for Dirichlet characters over cyclotomic fields

[pjbruin]

In SageMath 6.9.beta2:

sage: K.<z> = CyclotomicField(8)
sage: G = DirichletGroup(13, K)
sage: chi = G([z^2])
sage: chi.gauss_sum()
Traceback (most recent call last):
...
TypeError: Cannot coerce zeta52 into Cyclotomic Field of order 8 and degree 4

This arises because chi is actually defined over a smaller cyclotomic field. It can be fixed by letting gauss_sum() minimize the base ring.

Depends on #19056

Component: number theory

Keywords: Dirichlet character Gauss sum

Author: Peter Bruin

Branch/Commit: ad61452

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/19060

[pjbruin]

Commit: 6ea900b

[pjbruin]

Branch: u/pbruin/19060-gauss_sum_cyclotomic_field

[sagetrac-git]

Changed commit from 6ea900b to 95b5401

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

95b5401

Trac 19060: fix doctests

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

ca77b27	Trac 19056: implement gauss_sum and gauss_sum_numerical for Dirichlet characters over QQbar
c5f0a8a	Merge branch 'ticket/19056-gauss_sum_algebraic_field' into ticket/19060-gauss_sum_cyclotomic_field

[sagetrac-git]

Changed commit from 95b5401 to c5f0a8a

[pjbruin]

Dependencies: #19056

[pjbruin]

comment:4

Added #19056 as a dependency because it has positive review and there is a merge conflict.

[fchapoton]

comment:5

two failing doctest, problem of complex conjugate

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4236cdc

Trac 19060: make doctests less sensitive to numerical noise

[sagetrac-git]

Changed commit from c5f0a8a to 4236cdc

[fchapoton]

comment:8

Hmm. Precision seems too high to me.. Maybe 1e-15 would be better ?

and maybe replace the very small imaginary part in the doctest by 0.0*I ?

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4cd66fd

Trac 19060: increase doctest tolerance

[sagetrac-git]

Changed commit from 4236cdc to 4cd66fd

[pjbruin]

comment:10

Replying to @fchapoton:

Hmm. Precision seems too high to me.. Maybe 1e-15 would be better ?

You are right, done.

and maybe replace the very small imaginary part in the doctest by 0.0*I ?

I prefer to have the doctest output reflect the actual output (on some system) as closely as possible, and I have never seen 0.0*I in these tests (of course it would be the ideal answer).

[fchapoton]

comment:11

one failing doctest, need to relax the tolerance a little bit, see patchbot report

[sagetrac-git]

Changed commit from 4cd66fd to ad61452

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

ad61452

Trac 19060: further increase doctest tolerance

[fchapoton]

comment:14

ok, looks good to me

[vbraun]

comment:15

Reviewer name missing

[fchapoton]

Reviewer: Frédéric Chapoton

[fchapoton]

comment:16

sorry for that.

[vbraun]

Changed branch from u/pbruin/19060-gauss_sum_cyclotomic_field to ad61452