It's a bug! You checked against pari: the Sage function just calls pari anyway (try K.hilbert_symbol??) so this is a pari bug.
I checked with Magma and for the three symbols above the results were +1, -1, -1 so it is the middle one which is wrong. This requires a bug report to pari; they are very good about fixing bugs but it will take longer for the fix to find its way into Sage. John On 1 April 2014 14:19, AHaensch <[email protected]> wrote: > In general, it is known that for a field F and for a prime p, > (a,b)_p*(a,c)_p=(a,bc)_p for any a, b in F (cf O'meara 63:12). Here ( , )_p > denotes the Hilbert Symbol over F localized at p. > > 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 > > Why is this happening? I tried it using pari commands > (pari(K).nfhilbert(2*a,-1,p.pari_prime())) and I'm getting the same thing. > I'm afraid I'm making some very trivial error, but it's not obvious to me. > > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
