Exercise · ZW-LAW-OF-QUADRATIC-RECIPROCITY-008 · ★★★ Advanced
Does the congruence \(x^2\equiv 3\pmod{11}\) have a solution?
Choose one option.
Show answer
- D. Yes, because \(\left(\frac{3}{11}\right)=1\)
Explanation
Since \(3\equiv 11\equiv 3\pmod 4\), the product of the two symbols equals \(-1\). Since \(\left(\frac{11}{3}\right)=\left(\frac{2}{3}\right)=-1\), we get \(\left(\frac{3}{11}\right)=1\), so the congruence has a solution.
Review the concept: Law of quadratic reciprocity
The law of quadratic reciprocity relates the Legendre symbols \(\left(\frac{p}{q}\right)\) and \(\left(\frac{q}{p}\right)\) for distinct odd primes \(p\) and \(q\).
\[\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{(p-1)(q-1)}{4}}\]
More practice problems: law of quadratic reciprocity
- ZW-LAW-OF-QUADRATIC-RECIPROCITY-001 State the law of quadratic reciprocity for distinct odd primes p and q.
- ZW-LAW-OF-QUADRATIC-RECIPROCITY-002 According to the law, when is ((p)/(q))((q)/(p))=-1?
- ZW-LAW-OF-QUADRATIC-RECIPROCITY-003 Compute ((3)/(5))((5)/(3)).
- ZW-LAW-OF-QUADRATIC-RECIPROCITY-004 What is the supplement formula for ((-1)/(p)) when p is an odd prime?
- ZW-LAW-OF-QUADRATIC-RECIPROCITY-005 Compute ((2)/(7)).
- ZW-LAW-OF-QUADRATIC-RECIPROCITY-006 Compute ((-1)/(7)).
- ZW-LAW-OF-QUADRATIC-RECIPROCITY-007 Compute ((5)/(11)).
- ZW-LAW-OF-QUADRATIC-RECIPROCITY-009 By definition, when is ((a)/(p))=0?
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