Multiplicativity of the Legendre symbol

Builds on Legendre symbol Euler's criterion

Statement

Let $p$ be an odd prime, and let $a$ and $b$ be integers. Then $\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right) \left(\frac{b}{p}\right),$ where the symbols are Legendre symbols.

Remarks

The proof follows from Euler's criterion, because $(ab)^{(p-1)/2} = a^{(p-1)/2} b^{(p-1)/2}$ . In words: modulo $p$ , the product of two nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue. For example, $3$ and $5$ are nonresidues modulo $7$ , and $15 \equiv 1 \pmod 7$ is a residue. So the map $a \mapsto \left(\frac{a}{p}\right)$ is a completely multiplicative function of $a$ .

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Divisibility definition
Congruence modulo n definition
Prime number definition
Quadratic residue definition
Fermat's little theorem statement
Legendre symbol definition
Euler's criterion statement
Multiplicativity of the Legendre symbol statement

Tag	`0NW1JP7TBR`
URI	`m2://0NW1JP7TBR`
Kind	statement, theorem
Writing	W0 stub
MSC	11A15
Status	This copy of the site has no build data. Run `m2 status 0NW1JP7TBR`.
Uses	Legendre symbol, Euler's criterion

Multiplicativity of the Legendre symbol

Statement

Remarks

Reading path

Further reading

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