 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2009/c/c2/v/solution

By multiplicativity of the Legendre symbol and Euler criterion,

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

Combining this with part (iv) yields

(\frac{- 2}{p}) = {1, - 1, p \equiv 1, 3 (mod 8), p \equiv 5, 7 (mod 8) .

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