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I'm self-studying Ireland/Rosen, but question #5.36 doesn't make sense to me. It asks

Show that part (c) of Proposition 5.2.2 is true if $a$ is negative and $b$ is positive (both still odd).

Part (c) of the Proposition says if $a$ is odd and positive as well as $b$, then $$ \left(\frac{a}{b}\right)\left(\frac{b}{a}\right)=(-1)^{((a-1)/2)((b-1)/2)}. $$ By the way, these are Jacobi symbols, not Legendre symbols. But if $a$ is negative, then $\left(\frac{b}{a}\right)$ isn't defined, as far as I know, so I don't understand what I'm supposed to prove. Am I misunderstanding the question? Thanks.

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There is the Kronecker symbol.

A different approach is to define $\left( \frac{a}{-p} \right)$ the same way you do $\left( \frac{a}{p} \right)$; after all, $-p$ is a prime number, and "modulo $-p$" works out to the same thing as "modulo $p$".

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  • $\begingroup$ That makes it straightforward, thanks Hurkyl. $\endgroup$ – Noomi Holloway Jan 13 '13 at 0:58

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