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Year 2 of the mathematics and statistics course of the University of Oxford

Probability, statistics and options II course of the University of Oxford

Past exam of the Probability, statistics and options II course of the University of Oxford

2011

If $p$ divides $a$ or $b$, it divides their product and both sides of the proposed identity are zero. Otherwise use <Euler's criterion> and the ordinary exponent rule:
$$
\left(\frac{ab}p\right)\equiv(ab)^{(p-1)/2}
\equiv a^{(p-1)/2}b^{(p-1)/2}
\equiv\left(\frac ap\right)\left(\frac bp\right)\pmod p.
$$
Both sides are in $\{1,-1\}$, and these two values are distinct modulo the odd prime $p$. Hence the congruence is an equality:
$$
\boxed{\left(\frac{ab}p\right)=\left(\frac ap\right)\left(\frac bp\right).}
$$

Solution

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

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

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