Quadratic Mathematics | Part IB, 2003

(a) Suppose pp is an odd prime and aa an integer coprime to pp. Define the Legendre symbol (ap)\left(\frac{a}{p}\right) and state Euler's criterion.

(b) Compute (1p)\left(\frac{-1}{p}\right) and prove that

(abp)=(ap)(bp)\left(\frac{a b}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)

whenever aa and bb are coprime to pp.

(c) Let nn be any integer such that 1np21 \leqslant n \leqslant p-2. Let mm be the unique integer such that 1mp21 \leqslant m \leqslant p-2 and mn1(modp)m n \equiv 1(\bmod p). Prove that


(d) Find


Typos? Please submit corrections to this page on GitHub.