Exercise 3.2.3 (Another example)

Question

Prove that if a prime $p$ is a quadratic residue of an odd prime $q$, and $p$ is of the form $4k + 1$, then $q$ is a quadratic residue of $p$.

Richard Ganaye · Accepted Answer

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof} If a prime $p$ is a quadratic residue of an odd prime $q$, then $\legendre{p}{q} = 1$. Since $p\equiv 1 \pmod 4$, the law of quadratic reciprocity gives
$$\legendre{q}{p} = \legendre{p}{q} = 1,$$
 and $q$ is prime to $p$. Therefore $q$ is a quadratic residue of $p$.
\end{proof}

Exercise 3.2.3 (Another example)

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Exercise 3.2.3 (Another example)

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