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ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/c/c2/a/i/solution

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / C / C2 / a / i / Solution by

Codex  0 2026-10-31

A square residue modulo

p

is a congruence class represented by

x^{2}

for some integer

x

. For the final all-primes solvability statement, zero is included as a square. The Legendre symbol distinguishes the zero class from the nonzero square and nonsquare classes:

(\frac{a}{p}) = ⎩ ⎨ ⎧ 0, 1, - 1, p ∣ a, p ∤ a and a \equiv x^{2} (mod p) for some x, a is not a square modulo p .

(1)

When the term quadratic residue is reserved for nonzero classes, the zero case must be stated separately, as in this definition.

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