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On Euler's criterion

Published online by Cambridge University Press:  09 April 2009

Emma Lehmer
Emma Lehmer
Affiliation:
Berkeley, Calif., U.S.A.
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Euler's criterion states that if p is a prime then if and only if D is a k-th power residue of p.

However if (1) does not hold then (2)

where ακ is some κ-th root of unity modulo ρ.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 1 , August 1959 , pp. 64 - 70
Copyright
Copyright © Australian Mathematical Society 1959

References

[1]Whiteman, A. L., “Cyclotomy and Jacobsthal Sums”, Amer. Jour. of Math. 74 (1952), 89–99.CrossRefGoogle Scholar
[2]Jacobsthal, E., “Anwendungen einer Formel einer Formel aus der.Theorie der quadratischen Reste”, Dissertation (Berlin, 1906).Google Scholar
[3]Lehmer, , Emma, , “On the Number of Solutions of uk + D = w (mod p)”, Pacific Jour. of Math. 55 (1955), 103–118.CrossRefGoogle Scholar
[4]Lehmer, , Emma, , “The quintic character of 2 and 3”, Duke Math. Journal 18 (1951) 11–18.CrossRefGoogle Scholar