 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/ib/paper-1/17g/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

- 1

. It gives

(\frac{- 1}{p}) = (- 1)^{(p - 1) /2} = {1, - 1, p \equiv 1 (mod 4), p \equiv 3 (mod 4) .

For

p ∤ ab

, the same criterion gives

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

. Both sides are integers in

{1, - 1}

, which are distinct modulo an odd prime, so congruence implies equality. This proves multiplicativity of the Legendre symbol:

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

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