 Solution

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

p

divides

a

b

, it divides their product and both sides of the proposed identity are zero. Otherwise use Euler's criterion and the ordinary exponent rule:

(\frac{ab}{p}) \equiv (ab)^{(p - 1) /2} \equiv a^{(p - 1) /2} b^{(p - 1) /2} \equiv (\frac{a}{p}) (\frac{b}{p}) (mod p) .

Both sides are in

{1, - 1}

, and these two values are distinct modulo the odd prime

p

. Hence the congruence is an equality:

(\frac{ab}{p}) = (\frac{a}{p}) (\frac{b}{p}) .

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