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...
closed as too localized by Mark Sapir, Felipe Voloch, Peter Mueller, Charles, Derek Holt Jun 19 at 21:40
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$Z_p^*$ 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