The octic periodic polynomial

The coefficients and the discriminant of the octic period polynomial ψ 8 ( z ) {\psi _8}(z) are computed, where, for a prime p = 8 f + 1 p = 8f + 1 , ψ 8 ( z ) {\psi _8}(z) denotes the minimal polynomial over Q {\mathbf {Q}} of the period ( 1 / 8 ) ∑ n = 1 p − 1 exp ⁡ ( 2 π i n 8 / p ) (1/8)\sum \nolimits _{n = 1}^{p - 1} {\exp (2\pi i{n^8}/p)} . Also, the finite set of prime octic nonresidues ( mod p ) (\mod p) which divide integers represented by ψ 8 ( z ) {\psi _8}(z) is characterized.