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Euler's Criterion for Quintic Nonresidues
Published online by Cambridge University Press: 20 November 2018
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Let e be an integer ≧ 2, and p a prime = 1 (mod e). Euler's criterion states that for D ∊ Z,
(1.1)
if and only if D is an e-th power residue (mod p). If D is not an e-th power (mod p), one has
(1.2)
for some e-th root α(≠1) of unity (mod p). Sometimes expressions for roots of unity (mod p) can be given in terms of quadratic partitions of p. For example,
(1.3)
are the four distinct fourth roots of unity (mod p) for a prime p ≡ 1 (mod 4) in terms of a solution (a, b) of the diophantine system
(a, b unique), whereas for p ≡ 1 (mod 3), a solution (L, M) of the system
gives
(1.4)
as the three distinct cuberoots of unity (mod p).
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- Copyright © Canadian Mathematical Society 1985
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