#16043: Hilbert Symbol introduces bugs from Pari
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Reporter: annahaensch | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.2
Component: number fields | Resolution:
Keywords: | Merged in:
Authors: Anna Haensch | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by pbruin):
By the basic properties of the Hilbert symbol, (2, -1),,''p'',, = (2,
2),,''p'',, = (2, -2),,''p'',, = 1 for any ''p''. Hence if ''p'' is the
unique prime dividing 2 in ''K'', some other correct values are
* (''a'', 2),,''p'',, = -1
* (''a'', -2),,''p'',, = -1
* (''a'', -1),,''p'',, = 1
PARI also computes (''a'', 2),,''p'',, incorrectly. In the latest version
of PARI/GP (2.7.0, not yet in Sage), one has
{{{
gp > K=nfinit(x^2+5);
gp > p=idealfactor(K,2)[1,1];
gp > a=Mod(x,K.pol);
gp > nfhilbert(K,a,2,p)
%4 = 1 # should be -1
gp > nfhilbert(K,a,-2,p)
%5 = -1
gp > nfhilbert(K,a,-1,p)
%6 = 1
}}}
Have you reported the bug to the PARI developers? (See
`http://pari.math.u-bordeaux.fr/Bugs/Reporting.html` for how to do this.)
--
Ticket URL: <http://trac.sagemath.org/ticket/16043#comment:1>
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