3.I .9 F. 9 \mathrm{~F} \quad

Quadratic Mathematics | Part IB, 2002

Explain what is meant by a quadratic residue modulo an odd prime pp, and show that aa is a quadratic residue modulo pp if and only if a12(p1)1(modp)a^{\frac{1}{2}(p-1)} \equiv 1(\bmod p). Hence characterize the odd primes pp for which 1-1 is a quadratic residue.

State the law of quadratic reciprocity, and use it to determine whether 73 is a quadratic residue (mod 127).

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