Past exam of the options II course of the University of Oxford / 2012 / C / C2 / a / ii

Table of contents
- Solution ii

Solution

Euler's criterion states that

a^{(p - 1) /2} \equiv (\frac{a}{p}) (mod p) .

For

p ∤ a

this says that the power is

1

for a quadratic residue and

- 1

for a quadratic nonresidue. When

p ∣ a

, both sides are zero modulo

p

.

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