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Criteria for Biquadratic Residuacity Modulo a Prime p involving Quaternary Representations of p

Published online by Cambridge University Press:  20 November 2018

Kenneth S. Williams ,
Christian Friesen  and
Lawrence J. Howe
Kenneth S. Williams
Affiliation:
Carleton University, Ottawa, Ontario
Christian Friesen
Affiliation:
Carleton University, Ottawa, Ontario
Lawrence J. Howe
Affiliation:
Carleton University, Ottawa, Ontario
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In 1958, Hasse [10, p. 236], in connection with his work on the 2n-th power character of 2 in the cyclotomic field Q(exp(2πi/2n)), proved that for every prime p ≡ 1 (mod 16) the pair of equations

is always solvable in integers x, u, v, w. Later in 1972 Giudici, Muskat, and Robinson [7, p. 388] showed in their work on Brewer's character sums that Hasse's system is also solvable for primes p ≡ 7 (mod 16). Moreover they also showed [7, p. 345] that for primes p ≡ 1 (mod 5) the pair of equations

is solvable in integers x, u, v, w. In this paper we consider a pair of diophantine equations (involving a prime p and an integer m) which includes, the above two systems as the special cases when m = 2 and m = 5. The system is then used to give criteria for m to be a biquadratic residue modulo p.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 2 , 01 April 1985 , pp. 337 - 370
Copyright
Copyright © Canadian Mathematical Society 1985

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