 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2004/c/c2/b/solution

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

The latter symbol is

+ 1

when

p \equiv 1 (mod 3)

and

- 1

when

p \equiv 2 (mod 3)

. A prime greater than three occupies one of the classes

1, 5, 7, 11

modulo twelve. Combining its residues modulo three and four gives signs

+ 1, - 1, - 1, + 1

respectively. Hence

(\frac{3}{p}) = 1 for p \equiv \pm 1 (mod 12), (\frac{3}{p}) = - 1 for p \equiv \pm 5 (mod 12) .

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