#10464: m-th power residue symbol
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Reporter: katestange | Owner: davidloeffler
Type: enhancement | Status: needs_work
Priority: minor | Milestone: sage-4.6.1
Component: number fields | Keywords: cubic residue
Author: Katherine Stange | Upstream: N/A
Reviewer: | Merged:
Work_issues: docstring formatting |
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Comment(by fwclarke):
I think there are a couple of things which will make the ideal
`residue_symbol` function a little faster:
1. There should be no need to have the default `check=True` when
recursively applying the function to the prime factors of the ideal since
all the checks will have already been performed.
1. The final loop can be avoided by calculating a discrete log in the
residue field.
Thus I think the last few lines of the definition should read something
like:
{{{
if not self.is_prime():
return prod(Q.residue_symbol(e, m, check=False)**i for Q, i in
self.factor())
k = self.residue_field()
r = k(e)**((k.order() - 1)/m)
resroot = primroot**(rootorder/m)
from sage.groups.generic import discrete_log
j = discrete_log(k(resroot), k(r), ord=m)
return resroot**j
}}}
One other little thing. I think it is safer to write
`self.smallest_integer()` rather than `ZZ(self.gens()[0])`.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10464#comment:4>
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