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number > number theory > algebraic number theory > modular arithmetic > Euler's criterion

Preferred term

Euler's criterion

In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then
${\\displaystyle a^{\ frac {p-1}{2}}\\equiv {\egin{cases}\\;\\;\\,1{\\pmod {p}}&{\ ext{ if there is an integer }}x{\ ext{ such that }}x^{2}\\equiv a{\\pmod {p}},\\\\-1{\\pmod {p}}&{\ ext{ if there is no such integer.}}\\end{cases}}}$
Euler's criterion can be concisely reformulated using the Legendre symbol:
${\\displaystyle \\left({\ rac {a}{p}}\ ight)\\equiv a^{\ frac {p-1}{2}}{\\pmod {p}}.}$
The criterion dates from a 1748 paper by Leonhard Euler.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Euler%27s_criterion)

modular arithmetic

square number

critère d'Euler

French

URI

http://data.loterre.fr/ark:/67375/PSR-R4K99MK9-L

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