How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z\_p^\*$ contains an element $a$ of order $n$? No ideas at all...



For any n and some prime p there is an elemnet in Zp* of order n
$Z\_p^\*$ is of order $p-1$ so what you are really asking is for a prime in the arithmetic progression $n+1, 2n+1, 3n+1, \ldots$. This is true by Dirichlet's theorem, see <http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions>


