Olivier Ramare on Tue, 26 Aug 2003 12:33:25 +0200

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Re: Kronecker symbol

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.