 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2010/c/1/iii/solution

Write

s = (a / p)

. If

p ∣ a

, both sides of the proposed identity are zero. Otherwise

s = \pm 1

(\frac{s a}{p}) = (\frac{s}{p}) (\frac{a}{p}) = s^{(p - 1) /2} s = s^{(p + 1) /2} .

For the middle equality, Euler's congruence determines the sign exactly, since both the symbol and

s^{(p - 1) /2}

are in

{1, - 1}

and

p

is odd.

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