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Euler's Criterion for Quintic Nonresidues

Published online by Cambridge University Press:  20 November 2018

S. A. Katre  and
A. R. Rajwade
S. A. Katre
Affiliation:
Panjab University, Chandigarh, India
A. R. Rajwade
Affiliation:
Panjab University, Chandigarh, India
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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).

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 5 , 01 October 1985 , pp. 1008 - 1024
Copyright
Copyright © Canadian Mathematical Society 1985

References

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