 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2023/1/b/i/solution

For

p > 3

(\frac{3}{p}) = (- 1)^{(p - 1) /2} (\frac{p}{3}) .

1

when

p \equiv 1 (mod 3)

and

- 1

when

p \equiv 2 (mod 3)

. Combining it with the sign determined by

p (mod 4)

shows that a quadratic residue occurs precisely for

p \equiv 1 or 11 (mod 12) .

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