#16043: Hilbert Symbol introduces bugs from Pari
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Reporter: annahaensch | Owner:
Type: defect | Status:
Priority: major | needs_work
Component: number fields | Milestone: sage-6.2
Keywords: | Resolution:
Authors: Anna Haensch | Merged in:
Report Upstream: Reported upstream. No | Reviewers: Peter
feedback yet. | Bruin
Branch: | Work issues: add
Dependencies: #15767 | doctest
| Commit:
| Stopgaps:
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Changes (by jdemeyer):
* upstream: Fixed upstream, but not in a stable release. => Reported
upstream. No feedback yet.
Old description:
> For a field F and for a prime p, ( , )_p denotes the Hilbert Symbol over
> F localized at p. It is well known that (a,b)_p*(a,c)_p=(a,bc)_p for any
> a, b in F (cf O'meara 63:12). But I'm getting:
>
> {{{
> sage: K.<a>=NumberField(x^2+5)
> sage: p=K.primes_above(2)[0];p
> Fractional ideal (2, a + 1)
> sage: K.hilbert_symbol(2*a,-1,p)
> 1
> sage: K.hilbert_symbol(2*a,2,p)
> 1
> sage: K.hilbert_symbol(2*a,-2,p)
> -1
> }}}
>
> This is an upstream problem: [http://pari.math.u-bordeaux.fr/cgi-
> bin/bugreport.cgi?bug=1561]
New description:
For a field F and for a prime p, ( , )_p denotes the Hilbert Symbol over F
localized at p. It is well known that (a,b)_p*(a,c)_p=(a,bc)_p for any a,
b in F (cf O'meara 63:12). But I'm getting:
{{{
sage: K.<a>=NumberField(x^2+5)
sage: p=K.primes_above(2)[0];p
Fractional ideal (2, a + 1)
sage: K.hilbert_symbol(2*a,-1,p)
1
sage: K.hilbert_symbol(2*a,2,p)
1
sage: K.hilbert_symbol(2*a,-2,p)
-1
}}}
This is an upstream problem: [http://pari.math.u-bordeaux.fr/cgi-
bin/bugreport.cgi?bug=1561] and the follow-up
[http://pari.math.u-bordeaux.fr/cgi-bin/bugreport.cgi?bug=1569]
--
Comment:
Follow-up upstream bug: [http://pari.math.u-bordeaux.fr/cgi-
bin/bugreport.cgi?bug=1569]
--
Ticket URL: <http://trac.sagemath.org/ticket/16043#comment:10>
Sage <http://www.sagemath.org>
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