 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/c/c2/a/iii/solution

For distinct odd primes

p, q

(\frac{p}{q}) (\frac{q}{p}) = (- 1)^{(p - 1) (q - 1) /4} .

The usual supplementary laws are

(\frac{- 1}{p}) = (- 1)^{(p - 1) /2}, (\frac{2}{p}) = (- 1)^{(p^{2} - 1) /8} .

The first supplement also follows immediately by inserting

a = - 1

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