Exercise 3.2.2 (Example of quadratic reciprocity)

Question

Prove that if $p$ and $q$ are distinct primes of the form $4k+3$, and if $x^2 \equiv p \pmod q$ has no solution, then $x^2 \equiv q \pmod p$ has two solutions.

Richard Ganaye · Accepted Answer

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

\begin{proof} 
 If $p$ and $q$ are distinct primes of the form $4k+3$, by the law of quadratic reciprocity
 $$\legendre{p}{q} = - \legendre{q}{p}.$$
 If $x^2 \equiv p \pmod q$ has no solution, then $\legendre{q}{p} = -1$, thus $\legendre{p}{q} =1$, and $p, q $ are distinct primes, so $q 
ot \equiv 0 \pmod p$. Therefore $x^2 \equiv q \pmod p$ has two solutions.
\end{proof}

Exercise 3.2.2 (Example of quadratic reciprocity)

Answers

Comments

Exercise 3.2.2 (Example of quadratic reciprocity)

Answers

Comments

Add answer