Re: Kronecker symbol

[Olivier Ramare <ramare@agat.univ-lille1.fr>]

> Once you extend Legendre, I think you are out of the "a is a square mod 
> b" realm; for example, (-7|15)=1, but -7 (== 8) is not a square mod 15.

Of course, the kernel of this symbol is a subgroup of order 2
while the subgroup of squares is of order 2^{omega(q)}
(false if 2|q but but ...).

> I can't answer the second question, and a check of the source offers no 
> hints.  FWIW, I believe that quadratic (== real?) characters all have 
> the form (a|b) (due to Dirichlet, I think).

There's only one quadratic character modulo a prime power and then
extend the property by multiplicativity.

HTH,
Amities,
         Olivier