 Solution

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

The second supplementary law for the Legendre symbol is

(2/ p) = (- 1)^{(p^{2} - 1) /8}

. To multiply these signs, observe that for odd

a, b

\frac{a ^{2} b ^{2} - 1}{8} - \frac{a ^{2} - 1}{8} - \frac{b ^{2} - 1}{8} = \frac{( a ^{2} - 1 ) ( b ^{2} - 1 )}{8} .

Both factors in the numerator are divisible by

8

, so the right side is even. Applying this identity repeatedly to the prime factorization gives

(\frac{2}{Q}) = i \prod (- 1)^{(p_{i}^{2} - 1) /8} = (- 1)^{(Q^{2} - 1) /8} .

Again the convention at

Q = 1

agrees.

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