#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 | Keywords:
Merged in: | Authors: Anna Haensch
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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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
}}}
Performing the same calculations using pari commands yields the same
results -- unsurprising since hilbert_symbol just calls pari's
nfhilbert():
{{{
sage: p
Fractional ideal (2, a + 1)
sage: q=p.pari_prime()
sage: pari(K).nfhilbert(2*a,-1,p.pari_prime())
1
sage: pari(K).nfhilbert(2*a,2,p.pari_prime())
1
sage: pari(K).nfhilbert(2*a,-2,p.pari_prime())
-1
}}}
So it looks like pari is introducing the bug. Checking against Magma
gives 1,-1,-1 respectively for the three Hilbert symbols, which is what
you would expect to get.
--
Ticket URL: <http://trac.sagemath.org/ticket/16043>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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