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- Jan 26, 2012

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**Problem**: Let $p$ be an odd prime and $a\in\mathbb{Z}$. Define the Legendre symbol

\[\left(\frac{a}{p}\right)= \begin{cases}1 & \text{if $x^2\equiv a\pmod{p}$ has an integer solution} \\ 0 & \text{if $p\mid a$}\\ -1 & \text{if $x^2\equiv a\pmod{p}$ has no integer solution}\end{cases}\]

If $p$ is an odd prime and $a,b$ are two integers such that $(p,ab)=1$, then prove that

\[\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right) \left(\frac{b}{p}\right).\]

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**Hint**:

Let $p$ be an odd prime and $a$ an integer such that $(a,p)=1$. Then

\[a^{\frac{p-1}{2}}\equiv \left(\frac{a}{p}\right)\pmod{p}.\]

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