Question

Use Exercise 62 to show that if p is an odd prime and a and b are integers not divisible by p, then $\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$

Short answer

$\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$

Step-by-step solution

Step: 1

By exercise 62,

$\begin{array}{r}\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)\equiv a^{(p-1)/2}{b}^{(p-1)/2}=(ab{)}^{(p-1)/2}\\ \equiv \left(\frac{ab}{p}\right)(modp)\end{array}$

Hence$\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$