Past exam of the options II for mathematics and philosophy course of the University of Oxford / 2013 / C2 / a / v

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- Solution v

Solution

For distinct odd primes

p, q

, the quadratic reciprocity law is

(p / q) (q / p) = (- 1)^{\frac{p - 1}{2} \frac{q - 1}{2}} .

Thus the two symbols have opposite signs exactly when both primes are three modulo four; otherwise they agree. Distinctness excludes zero symbols.

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