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Year 2 of the mathematics course of the University of Oxford

Options II course of the University of Oxford

Past exam of the options II course of the University of Oxford

2011

If $p$ divides $a$ or $b$, both sides of the claimed <multiplicativity> identity are zero. Otherwise the <Euler criterion> gives
$$
\left(\frac{ab}p\right)\equiv (ab)^{(p-1)/2}
\equiv a^{(p-1)/2}b^{(p-1)/2}
\equiv\left(\frac ap\right)\left(\frac bp\right)\pmod p.
$$
Each side is $1$ or $-1$. Since $p$ is odd, these two integers represent different residue classes. Hence the <modular congruence> gives equality:
$$
\boxed{\left(\frac{ab}p\right)=\left(\frac ap\right)\left(\frac bp\right).}
$$

Solution

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

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

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

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