Euler's Criterion for Quintic Nonresidues

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 systemgives(1.4)as the three distinct cuberoots of unity (mod p).