PlanetMath: values of the Legendre symbol

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (47)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Copyright
About

Entry average rating: No information on entry rating

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:

$\begin{displaymath} \left(\frac{-1}{p}\right) = \begin{cases} 1 &\text{if }p \equiv 1 \mod 4\ -1 &\text{if }p \equiv 3 \mod 4. \end{cases}\end{displaymath}$
$\begin{displaymath} \left(\frac{2}{p}\right) = \begin{cases} 1 &\text{if }p\equiv \pm 1 \mod 8\ -1 &\text{if }p\equiv 3,5 \mod 8. \end{cases}\end{displaymath}$
$\begin{displaymath} \left(\frac{3}{p}\right) = \begin{cases} 1 &\text{if }p\equiv \pm 1 \mod 12\ -1 &\text{otherwise.} \end{cases}\end{displaymath}$
$\begin{displaymath} \left(\frac{5}{p}\right) = \begin{cases} 1 &\text{if }p\equiv \pm 1 \mod 5\ -1 &\text{if }p\equiv 2,3 \mod 5. \end{cases}\end{displaymath}$

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, $$\left(\frac{5}{p}\right)=\left(\frac{p}{5}\right)$$ and the only quadratic residues modulo $5$ are $\pm 1 \mod 5$ . $\qedsymbol$

"values of the Legendre symbol" is owned by alozano.

(view preamble | get metadata)

See Also: cases when minus one is a quadratic residue, quadratic character of 2

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: quadratic residues, quadratic reciprocity, proof, Legendre symbol, prime, odd, integer
There is 1 reference to this entry.

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 1335 times total.

Classification:

AMS MSC:

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

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact