Exercise · ZW-LAW-OF-QUADRATIC-RECIPROCITY-004 · ★☆☆ Basic
What is the supplement formula for \(\left(\frac{-1}{p}\right)\) when \(p\) is an odd prime?
Choose one option.
Show answer
- D. \((-1)^{(p-1)/2}\)
Explanation
The symbol \(\left(\frac{-1}{p}\right)\) equals \((-1)^{(p-1)/2}\), which depends on \(p\) modulo 4. The option with \((p^2-1)/8\) belongs to the symbol of \(2\), not \(-1\).
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-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-008 Does the congruence x^2≡ 3pmod11 have a solution?
- ZW-LAW-OF-QUADRATIC-RECIPROCITY-009 By definition, when is ((a)/(p))=0?
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