#9334: hilbert symbols!!!
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Reporter: aly.deines | Owner: davidloeffler
Type: enhancement | Status: needs_work
Priority: minor | Milestone: sage-5.0
Component: number fields | Keywords: hilbert symbol
Author: aly.deines | Upstream: N/A
Reviewer: David Loeffler, John Cremona | Merged:
Work_issues: ReST formatting issues, and more |
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Changes (by cremona):
* cc: mstreng (added)
* reviewer: => David Loeffler, John Cremona
* work_issues: ReST formatting issues => ReST formatting issues, and
more
* milestone: sage-wishlist => sage-5.0
Comment:
I generally agree with David's points. This code will be very useful for a
topic begin done at SD23 (solving conics over various fields) so I am keen
to get this in (suitably modified).
In generalized_legendre_symbol: (1) test P for primality first, before
trying to construct its residue field. (2) instead of K(2).valuation(P)
just test that n is odd. (3) don't raise run-time errors, make them
ValueErrors?. (4) make the return types consistent: you return either +1
in k or -1 as a python int. I would return a Sage integer in either case.
(5) you do not test if P divides self. If so, return 0 (as a Sage
integer)>
Why are generalized_hilbert_symbol and _legendre_symbol in
sage/rings/arith.py? I would put them both in number_fields -- where you
put the even one in fact.
In generalized_even_hilbert_symbol you define but do not use iprime, so
delete it. And do the simple calculation to get the coefficients of
jprime**2 so you don't need to construct the quaternion algebra. (You can
leave in a comment about that).
_voight_alg_6_2 has some ^ symbols which should be **. Check for others.
Do what David said about uniformizer -- just call the existing function.
Sort out the solve function.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9334#comment:8>
Sage <http://www.sagemath.org>
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