quaternion ideals' .scale() incorrectly copies cached left and right orders

[yyyyx4]

The method QuaternionFractionalIdeal_rational.scale() always copies over the existing left and right orders of self to the scaled ideal. This is incorrect when the scaling factor does not lie in the center of the algebra, as demonstrated by this example:

sage: Quat.<i,j,k> = QuaternionAlgebra(-1,-19)
sage: a = 1+2*i+3*j+4*k
sage: I = Quat.maximal_order().unit_ideal()
sage: I.right_order()
Order of Quaternion Algebra (-1, -19) with base ring Rational Field with basis (1/2 + 1/2*j, 1/2*i + 1/2*k, j, k)
sage: I.scale(a).right_order()
Order of Quaternion Algebra (-1, -19) with base ring Rational Field with basis (1/2 + 1/2*j, 1/2*i + 1/2*k, j, k)
sage: J = Quat.ideal(I.scale(a).basis(), check=False)
sage: J == I.scale(a)
True
sage: J.right_order()
Order of Quaternion Algebra (-1, -19) with base ring Rational Field with basis (1/2 + 1/10*j + 109/5*k, 1/48*i + 1/120*j + 8411/240*k, 1/5*j + 218/5*k, 120*k)

The patch fixes this by only copying over the cached orders when scaling on the other side, or when scaling by an element of QQ.

CC: @pjbruin

Component: algebra

Stopgaps: mathematically_wrong

Author: Lorenz Panny

Branch/Commit: 4310b6c

Reviewer: Michael Orlitzky

Issue created by migration from https://trac.sagemath.org/ticket/32726

[yyyyx4]

Branch: public/32726

[yyyyx4]

Author: Lorenz Panny

[yyyyx4]

Commit: ef98f79

[yyyyx4]

comment:2

Bumping priority since this bug can lead to mathematical errors.

[mkoeppe]

comment:3

Stalled in needs_review or needs_info; likely won't make it into Sage 9.5.

[sagetrac-git]

Changed commit from ef98f79 to 4310b6c

[sagetrac-git]

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

4310b6c

Merge tag '9.5.rc2' into public/32726

[orlitzky]

comment:5

This looks OK to me but I'm not an expert so I have two questions:

1. I guess scaling on the right doesn't affect the left order, and vice-versa?
2. Is it OK to assume that the base ring is QQ?

[yyyyx4]

comment:6

Thanks for having a look.

1. Yes; the left order of I is {x in the algebra | x·I ⊆ I}. Scaling happens element-wise, so these inclusions are preserved under right scaling. Similarly on the other side.
2. Yes; the class QuaternionFractionalIdeal_rational is specific to QQ.

[orlitzky]

comment:7

LGTM then, thanks.

[orlitzky]

Reviewer: Michael Orlitzky

[vbraun]

Changed branch from public/32726 to 4310b6c