- #1
Evgeny.Makarov
Gold Member
MHB
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This is a basic question about the behavior of $a^k$ for $a\in\mathbb{Z}/n\mathbb{Z}$ and $k=1,2,\dots$. I know that Fermat's little theorem says $a^{p-1}=1$ in $\mathbb{Z}/p\mathbb{Z}$ for prime $p$; thus, the sequence $1,a,a^2,a^3,\dots$ is periodic with period at most $p-1$. More generally, by Euler's theorem $a^{\varphi(n)}=1$ in $\mathbb{Z}/n\mathbb{Z}$ if $(a,n)=1$, so $1,a,a^2,a^3,\dots$ is periodic with period at most $\varphi(n)$. But what happens when $(a,n)\ne1$?
The motivation for the question is to find out the general form of a polynomial in $\mathbb{Z}/n\mathbb{Z}[x]$.
The motivation for the question is to find out the general form of a polynomial in $\mathbb{Z}/n\mathbb{Z}[x]$.