Hilbert Symbol Over Number Fields Not Multiplicative

[AHaensch]

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. 

[John Cremona]

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

> --
> 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 sage-support...@googlegroups.com.
> To post to this group, send email to sage-s...@googlegroups.com.
> Visit this group at http://groups.google.com/group/sage-support.
> For more options, visit https://groups.google.com/d/optout.