 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2021/1/b/i/solution

The supplementary law for the Legendre symbol is

(- 1/ p) = (- 1)^{(p - 1) /2}

p

a, b

\frac{ab - 1}{2} = \frac{a - 1}{2} + \frac{b - 1}{2} + \frac{( a - 1 ) ( b - 1 )}{2},

and the last term is even. Induction over the prime factors of

Q

therefore gives

\sum_{i} (p_{i} - 1) /2 \equiv (Q - 1) /2 (mod 2)

. Consequently

(\frac{- 1}{Q}) = i \prod (- 1)^{(p_{i} - 1) /2} = (- 1)^{(Q - 1) /2} .

For

Q = 1

the exponent is zero.

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