Let $f(x)=x^{5}+ax^{3}+bx^{2}+cx+d\in\mathbb{Z[}x]$ 

have Galois group $\mathbb{Z}/5\mathbb{Z}$. 

The set of primes $q$ for which $f(x) \equiv(x+r)^{5}$ 

$(\operatorname{mod}q)$ for some $r\in\mathbb{Z}$ 

is determined.

\ The algorithm of Kobayashi and Nakagawa 

for solving the quintic equation

$x^{5}+ax^{3}+bx^{2}+cx+d=0$ is discussed 

in relation to this determination.