 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-3/11h/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

p

(\frac{x}{p}) = ⎩ ⎨ ⎧ 0, 1, - 1, p ∣ x, x \neq \equiv 0 (mod p) is a <quadratic residue>, x is a <quadratic nonresidue> .

p ∤ x

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

Both factors belong to

{1, - 1}

, so they are equal. Thus

(x^{- 1} / p) = (x / p)

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