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[parent] values of the Legendre symbol (Theorem)

For an integer $ a$ and an odd prime $ p$, let $ \displaystyle \left(\frac{a}{p}\right)$ be the Legendre symbol.

Theorem 1   Let $ p$ be an odd prime. The Legendre symbol takes the following values:
  1. \begin{displaymath} \left(\frac{-1}{p}\right) = \begin{cases} 1 &\text{if }p \equiv 1 \mod 4\ -1 &\text{if }p \equiv 3 \mod 4. \end{cases}\end{displaymath}
  2. \begin{displaymath} \left(\frac{2}{p}\right) = \begin{cases} 1 &\text{if }p\equiv \pm 1 \mod 8\ -1 &\text{if }p\equiv 3,5 \mod 8. \end{cases}\end{displaymath}
  3. \begin{displaymath} \left(\frac{3}{p}\right) = \begin{cases} 1 &\text{if }p\equiv \pm 1 \mod 12\ -1 &\text{otherwise.} \end{cases}\end{displaymath}
  4. \begin{displaymath} \left(\frac{5}{p}\right) = \begin{cases} 1 &\text{if }p\equiv \pm 1 \mod 5\ -1 &\text{if }p\equiv 2,3 \mod 5. \end{cases}\end{displaymath}
Proof. For a proof of (1), see this entry. Part (2) is proved in this entry. For parts (3), (4) and (5), we use quadratic reciprocity. For example,
$\displaystyle \left(\frac{5}{p}\right)=\left(\frac{p}{5}\right)$
and the only quadratic residues modulo $ 5$ are $ \pm 1 \mod 5$. $ \qedsymbol$



"values of the Legendre symbol" is owned by alozano.
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See Also: cases when minus one is a quadratic residue, quadratic character of 2


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Cross-references: quadratic residues, quadratic reciprocity, Legendre symbol, prime, odd, integer
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This is version 2 of values of the Legendre symbol, born on 2006-10-06, modified 2006-10-06.
Object id is 8425, canonical name is ValuesOfTheLegendreSymbol.
Accessed 741 times total.

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AMS MSC: 11-00 (Number theory :: General reference works )
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