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

Williams, Kenneth S. ; Friesen, Christian ; Howe, Lawrence J.

Canadian journal of mathematics, Tome 37 (1985) no. 2, pp. 337-370

Résumé

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.

Détail

DOI : 10.4153/CJM-1985-021-1

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